# Unit conversions in chemistry

##### Intros
###### Lessons
1. Introduction to unit conversions
2. Units and calculations in chemistry.
3. What is a unit conversion?
4. Unit conversion method: Walkthrough
##### Examples
###### Lessons
1. Apply the conversion factor method to simple calculations.
Use the unit conversion method to answer the following problems.
1. If a car can travel 75 kilometres in 1 hour, how far can it travel in 4.5 hours?
2. An electronics store has an offer that sells two TVs for $430. How much would 8 TVs cost using this deal? 3. At a market, a man traded 8 apples for 14 oranges. How many oranges can he get if he traded 22 apples at this rate? 2. Apply the conversion factor method to chemistry-related calculations. Use the unit conversion method to answer the following problems. 1. If a container holds 3.5 dozen oxygen atoms, how many oxygen atoms in total will 3 full containers have in them? 2. If one mole of oxygen gas has a mass of 32 g, what is the mass of 3.5 moles of oxygen gas? 3. If one molecule of white phosphorus has 4 atoms of phosphorus in it, how many molecules of white phosphorus would be needed to have 72 atoms of phosphorus? 4. If one mole of hydrogen gas fills up 22.4 L in a gas canister, how many moles of hydrogen gas would fill up a gas canister 180 L in size? 3. Apply the conversion factor method to chemistry-related calculations with SI units. Use the unit conversion method to answer the following problems. 1. 4.7 moles of carbon dioxide gas has a mass of 206.8 g. What is the mass of 1 mole of carbon dioxide gas? 2. Gold has a density of 19.3 grams per millilitre (g/mL), what would the volume be of 62 grams of gold? 3. If an acid has a concentration of 3 moles per litre, how many litres of acid would I need to have 1.8 moles of acid? ###### Topic Notes ## Introduction to Unit Conversions in Chemistry Welcome to our lesson on chem conversions, a fundamental skill in chemistry that's crucial for accurate calculations and problem-solving. Chemistry unit conversions are essential for translating between different measurement systems and scales, allowing scientists to work with diverse data sets and communicate results effectively. Whether you're balancing chemical equations, analyzing chemical reactions, or preparing solutions, mastering the unit conversion method is key to success in chemistry. This video lesson will guide you through the basics of chemistry unit conversions, providing clear examples and step-by-step instructions. You'll learn how to use conversion factors, dimensional analysis, and the factor-label method to solve a variety of problems. By the end of this lesson, you'll have a solid foundation in this critical skill, enabling you to tackle more complex chemical calculations with confidence. Let's dive in and explore the world of unit conversions in chemistry! ## Understanding the Basics of Unit Conversions Welcome to the world of chemistry unit conversions! If you've ever wondered how to do unit conversions in chemistry, you're in the right place. Let's dive into this essential skill that every chemistry student needs to master. At its core, unit conversion is all about expressing the same quantity in different units. For example, you might need to convert grams to moles, or liters to milliliters. This is where conversion factors come into play. A conversion factor is a ratio that shows the relationship between two different units. It's like a magic key that unlocks the door between different measurement systems. Why are conversion factors so important? Well, they're the backbone of chemistry unit conversion. They allow us to move seamlessly between different units, ensuring that our calculations are accurate and meaningful. Without conversion factors, we'd be stuck in a world where we couldn't compare apples to oranges or in this case, grams to moles! Now, let's break down a typical conversion problem into its three key components: 1. Conversion Statement: This is the heart of your problem. It tells you what you're starting with and where you need to end up. For example, "Convert 5 grams of sodium to moles." 2. Initial Amount: This is your starting point. In our example, it's 5 grams of sodium. 3. Final Amount: This is where you want to end up. In our case, it's moles of sodium. Let's look at a simple example to illustrate these concepts. Suppose we want to convert 10 meters to centimeters. Our conversion statement is "Convert 10 meters to centimeters." The initial amount is 10 meters, and our final amount will be in centimeters. We know that 1 meter = 100 centimeters, so our conversion factor is 100 cm/1 m. To solve this, we multiply our initial amount by the conversion factor: 10 m × (100 cm / 1 m) = 1000 cm See how the units of meters cancel out, leaving us with centimeters? That's the beauty of conversion factors! In chemistry, you'll often encounter more complex conversions. For instance, you might need to convert grams to moles using the molar mass of a substance. The principle remains the same identify your conversion factor (in this case, molar mass), set up your equation, and let the units guide you to the correct answer. Remember, practice makes perfect when it comes to chemistry unit conversion. Don't be discouraged if it seems tricky at first. With time and practice, you'll find yourself converting units like a pro! As you continue your chemistry journey, you'll discover that unit conversions are a fundamental skill that underpins many more complex concepts. Whether you're balancing equations, calculating concentrations, or working with gas laws, your ability to convert units will be invaluable. So, the next time you're faced with a chemistry problem that requires unit conversion, take a deep breath and remember these basics. Identify your conversion statement, initial amount, and final amount. Find the appropriate conversion factor, set up your equation, and let the units guide you to the solution. With these tools in your chemistry toolkit, you'll be well-equipped to tackle a wide range of problems and deepen your understanding of this fascinating subject. ## The Unit Conversion Method: Step-by-Step Approach The unit conversion method is a fundamental skill in chemistry and other sciences, allowing us to convert between different units of measurement. This step-by-step guide will walk you through the process, using the eggs and boxes example from the video to demonstrate how to do conversions in chemistry effectively. Step 1: Identify the given information and the desired result Before beginning any conversion, clearly identify what information you have and what you need to find. In our example, we might start with 24 eggs and want to know how many boxes this represents. Step 2: Set up the equation Start with the given quantity and unit on one side of the equation. In this case, we'd begin with 24 eggs. Step 3: Choose the correct conversion factor A conversion factor is a ratio that relates two different units. In our example, if we know that one box contains 12 eggs, our conversion factor would be 1 box / 12 eggs. Step 4: Write out the full equation Place your conversion factor next to the initial quantity, creating a multiplication problem. It should look like this: 24 eggs × (1 box / 12 eggs) Step 5: Cancel out common units This is a crucial step in chem unit conversions. The unit 'eggs' appears in both the numerator and denominator, so it can be canceled out. Cross out 'eggs' in both places. Step 6: Perform the calculation Now that the units are canceled, you can perform the mathematical operation: 24 × (1 / 12) = 2 boxes Step 7: Check your answer Always verify that your result makes sense. In this case, 2 boxes of 12 eggs each indeed equals 24 eggs. The importance of unit cancellation in problem-solving cannot be overstated. It serves as a built-in check to ensure you're using the correct conversion factors and helps prevent errors in your calculations. When units cancel properly, you can be confident that your conversion is set up correctly. Let's consider another example to reinforce the unit conversion method. Suppose you need to convert between units like kilometers and meters. Here's how you'd apply the steps: 1. Given: 5 kilometers; Need to find: equivalent in meters 2. Set up the equation: Start with 5 km 3. Conversion factor: 1 km = 1000 m, so use 1000 m / 1 km 4. Full equation: 5 km × (1000 m / 1 km) 5. Cancel units: The 'km' units cancel out 6. Calculate: 5 × 1000 = 5000 meters This method is versatile and can be applied to more complex conversions in chemistry. For instance, when converting between moles and grams, or when dealing with concentration units like molarity. The key is to always use appropriate conversion factors and pay close attention to unit cancellation. Practice is essential to mastering the unit conversion method. Start with simple conversions and gradually work your way up to more complex problems. Remember, in chemistry, units are just as important as the numerical values they accompany. Proper unit management through conversion ensures that your calculations are not only mathematically correct but also scientifically meaningful. By following these steps and understanding the principles behind unit conversion, you'll be well-equipped to handle a wide range of problems in chemistry and beyond. Whether you're balancing chemical equations, calculating molar masses, or solving stoichiometry problems, the unit conversion method will be an invaluable tool in your problem-solving arsenal. ## Applying Unit Conversions to Chemistry Problems Unit conversions are a fundamental skill in chemistry, allowing us to navigate between different measurements and quantities. Just like we saw with the egg and dollar example, chemistry problems often require multiple conversions to reach the desired result. Let's explore how to apply these techniques to chemistry-specific problems, focusing on common units like grams, moles, and liters. One of the most frequent conversions in chemistry involves moving between grams and moles. The mole is a crucial unit in chemistry, representing 6.022 x 10^23 particles of a substance. To convert between grams and moles, we use the substance's molar mass. For example, let's convert 50 grams of sodium (Na) to moles: 1. Identify the molar mass of Na: 22.99 g/mol 2. Set up the conversion: 50 g Na × (1 mol Na / 22.99 g Na) 3. Calculate: 50 / 22.99 = 2.17 mol Na This process works both ways. If we needed to convert moles to grams, we'd simply flip the fraction: (22.99 g Na / 1 mol Na). Now, let's tackle a more complex problem involving multiple conversions, similar to our egg and dollar example. Suppose we need to convert 5 liters of hydrogen gas at standard temperature and pressure (STP) to grams. Step 1: Convert liters to moles using the molar volume of a gas at STP (22.4 L/mol) 5 L H2 × (1 mol H2 / 22.4 L H2) = 0.223 mol H2 Step 2: Convert moles to grams using the molar mass of H2 (2.016 g/mol) 0.223 mol H2 × (2.016 g H2 / 1 mol H2) = 0.450 g H2 By chaining these conversions, we've solved a problem that initially seemed complex. This method of breaking down conversions into manageable steps is key to mastering chemistry calculations. Let's explore another common scenario: converting concentration units. Imagine we have a solution with a concentration of 0.5 M (moles per liter) of NaCl, and we need to express this in grams per liter. Step 1: Understand that 0.5 M means 0.5 moles of NaCl per liter of solution Step 2: Convert moles of NaCl to grams using its molar mass (58.44 g/mol) 0.5 mol NaCl/L × (58.44 g NaCl / 1 mol NaCl) = 29.22 g NaCl/L This example demonstrates how unit conversions can help us translate between different ways of expressing concentration, a crucial skill in chemistry lab work and calculations. As you practice these conversions, remember that each step is an opportunity to check your work. Always write out your units and ensure they cancel out correctly. This not only helps prevent errors but also builds your confidence in handling complex problems. Don't be discouraged if these conversions seem challenging at first. Like any skill, mastery comes with practice. Start with simple conversions and gradually work your way up to more complex, multi-step problems. With time, you'll find yourself navigating through chemistry calculations with ease. Remember, the key to success in chemistry conversions is to approach each problem systematically: 1. Identify the given information and the desired result 2. Determine the necessary conversion steps 3. Write out each step, clearly showing your units 4. Perform the calculations, canceling units as you go 5. Check your final answer for reasonableness By following these steps and practicing regularly, you'll develop a strong foundation in converting concentration units. This skill will serve you well in your chemistry studies and beyond. ## Common Pitfalls and Tips for Success in Chemistry Unit Conversions Unit conversions are a fundamental skill in chemistry, but they can be tricky for many students. Let's explore some common mistakes and provide helpful tips to master this essential technique. Remember, making errors is a natural part of learning, so don't get discouraged! One of the most frequent mistakes in chemistry unit conversion is forgetting to write out units. Always include units throughout your calculations. This simple step helps you visualize the conversion process and spot errors more easily. For example, when converting grams to moles, write "g" and "mol" at each step. Another common pitfall is improper unit cancellation. When setting up your conversion, ensure that units cancel out correctly. Arrange your conversion factors so that unwanted units appear in both the numerator and denominator, allowing them to cancel. This technique is crucial for arriving at the correct final unit. Students often struggle with selecting the appropriate conversion factor. To avoid this, create a "roadmap" of your conversion before starting calculations. Identify your starting unit and desired final unit, then determine the necessary steps to get there. This planning phase can prevent many errors and save time in the long run. Misplacing decimal points is another frequent error in chemistry unit conversions. Always double-check your decimal placement, especially when dealing with scientific notation. A misplaced decimal can lead to answers that are off by orders of magnitude. To improve your accuracy, develop the habit of estimating reasonable answers before calculating. This practice helps you catch major errors quickly. For instance, if you're converting 100 grams to kilograms, you know the answer should be less than 1 kg. If your calculation yields 10 kg, you'll immediately recognize something's amiss. When performing multi-step conversions, break the process down into smaller, manageable steps. This approach reduces the likelihood of errors and makes it easier to track your work. It's also helpful for identifying where a mistake might have occurred if your final answer seems off. Consistency in significant figures is crucial in chemistry. Many students forget to apply significant figure rules in their unit conversions. Remember to carry all digits through intermediate steps and round only at the end of your calculation. To further enhance your unit conversion skills, practice regularly with a variety of problems. The more you work with different types of conversions, the more comfortable and proficient you'll become. Don't hesitate to seek help from your instructor or classmates if you're struggling with a particular concept. Lastly, always review your work. After completing a conversion, take a moment to go through your steps again. Check for proper unit cancellation, correct multiplication and division, and reasonable final answers. This final review can catch many overlooked errors and improve your overall accuracy in chemistry unit conversions. Remember, mastering unit conversions in chemistry takes time and practice. Be patient with yourself and celebrate your progress along the way. With consistent effort and attention to detail, you'll soon find these conversions becoming second nature. Keep up the great work! To further enhance your unit conversion skills, practice regularly with a variety of problems. The more you work with different types of conversions, the more comfortable and proficient you'll become. Don't hesitate to seek help from your instructor or classmates if you're struggling with a particular concept. Consistency in significant figures is crucial in chemistry. Many students forget to apply significant figure rules in their unit conversions. Remember to carry all digits through intermediate steps and round only at the end of your calculation. ## Practice Problems and Real-World Applications Ready to put your unit conversion skills to the test? Let's dive into some practice problems that will sharpen your abilities and show you how these skills apply in various real-world scenarios. Remember, mastering unit conversions is crucial not just for chemistry, but for many other fields as well! ### Problem 1: Simple Conversion Convert 5.2 kilometers to meters. Solution: 1 km = 1000 m 5.2 km × (1000 m / 1 km) = 5200 m ### Problem 2: Multi-Step Conversion Convert 3.5 hours to seconds. Solution: Step 1: Convert hours to minutes 3.5 hours × (60 minutes / 1 hour) = 210 minutes Step 2: Convert minutes to seconds 210 minutes × (60 seconds / 1 minute) = 12,600 seconds ### Problem 3: Chemistry-Specific Conversion Convert 0.025 moles of NaCl to grams. (Molar mass of NaCl = 58.44 g/mol) Solution: 0.025 mol NaCl × (58.44 g NaCl / 1 mol NaCl) = 1.461 g NaCl ### Problem 4: Real-World Scenario A recipe calls for 250 mL of milk. Your measuring cup only shows cups. How many cups of milk should you use? (1 cup = 236.588 mL) Solution: 250 mL × (1 cup / 236.588 mL) = 1.057 cups Round to 1.06 cups for practical use in cooking. ### Problem 5: Complex Conversion Convert 65 miles per hour to meters per second. Solution: Step 1: Convert miles to meters 65 miles × (1609.34 meters / 1 mile) = 104,607.1 meters Step 2: Convert hours to seconds 1 hour × (3600 seconds / 1 hour) = 3600 seconds Step 3: Divide meters by seconds 104,607.1 meters / 3600 seconds = 29.06 m/s Great job working through these problems! As you can see, unit conversion skills are essential in various contexts. In chemistry, you'll frequently convert between moles, grams, and liters. But these skills extend far beyond the lab: • Engineering: Converting between different units of force, pressure, and energy. • Medicine: Calculating correct dosages by converting between different units of measurement. • Cooking: Adjusting recipes by converting between different volume and mass units. • Travel: Converting between miles and kilometers, or different currencies. • Sports: Converting speeds from mph to m/s, or distances from yards to meters. The key to mastering unit conversions is practice and understanding the relationships between different units. Don't be discouraged if you find some conversions challenging at first it's a skill that improves with time and effort. Remember these tips: 1. Always write out your units and cancel them out to ensure you're on the right track. 2. Use dimensional analysis to set up your conversions step-by-step ## Conclusion: Mastering Unit Conversions in Chemistry In this article, we've explored the crucial skill of chemistry unit conversion. We've covered the importance of dimensional analysis, the use of conversion factors, and strategies for tackling complex multi-step conversions. Mastering these chem conversions is essential for success in chemistry and beyond. Remember, practice makes perfect! We encourage you to review the introductory video and work through additional problems regularly to build your confidence and speed. Unit conversion mastery is within your reach. If you're struggling, don't hesitate to seek help from your instructor or classmates. Challenge yourself with more advanced problems to further hone your skills. By dedicating time to mastering this fundamental skill, you're setting yourself up for success in all areas of chemistry. Keep pushing forward, and soon you'll find unit conversions becoming second nature. Your efforts will pay off in your chemistry studies and future scientific endeavors. Ready to take your skills to the next level? Try our additional practice sets and keep that chemistry momentum going! ### Introduction to Unit Conversions Units and calculations in chemistry. #### Step 1: Understanding the Importance of Unit Conversions Unit conversions are essential in chemistry because they allow us to work with different units of measurement seamlessly. Whether you're dealing with mass, volume, concentration, or any other measurable quantity, being able to convert between units is crucial for accurate calculations. This skill is not only valuable in chemistry but also in other sciences and everyday life scenarios where different units of measurement are used. #### Step 2: Common Units of Measurement in Chemistry In chemistry, you will encounter various units of measurement. Some of the most common ones include: • Grams (g): Used to measure mass or the amount of a substance. • Litres (L): Used to measure volume, which can apply to both liquids and gases. • Moles (mol): A unit that measures the amount of atoms or molecules in a substance. Understanding these units and how to convert between them is fundamental for solving chemistry problems. #### Step 3: The Unit Conversion Method The unit conversion method is a systematic approach to converting one unit of measurement to another. This method is preferred because it is self-correcting, meaning that if you make a mistake, it will be evident in your calculations. The basic idea is to use conversion factors, which are ratios that express how many of one unit are equal to another unit. For example, if you know that 1 minute is equal to 60 seconds, you can use this conversion factor to convert between minutes and seconds. #### Step 4: Practical Examples of Unit Conversions Let's look at some practical examples to strengthen our understanding of unit conversions: • Example 1: Converting time units. If you have 60 seconds, you can convert this to minutes by using the conversion factor 1 minute = 60 seconds. Therefore, 60 seconds is equal to 1 minute. • Example 2: Converting volume units. If you have 1 litre of a liquid, you might need to convert this to millilitres. Knowing that 1 litre = 1000 millilitres, you can easily make this conversion. • Example 3: Converting mass units. If you have 1000 grams of a substance, you might need to convert this to kilograms. Since 1 kilogram = 1000 grams, you can convert 1000 grams to 1 kilogram. These examples illustrate how unit conversions are applied in various scenarios, making it easier to work with different units of measurement. #### Step 5: Applying Unit Conversions in Chemistry Calculations In chemistry, unit conversions are frequently used in calculations involving moles, mass, volume, and concentration. For instance, when calculating the number of moles of a substance, you might need to convert the given mass in grams to moles using the molar mass of the substance. Similarly, when working with solutions, you might need to convert between different units of concentration, such as molarity (moles per litre) and mass percent. By mastering unit conversions, you will be better equipped to handle complex chemistry problems and ensure the accuracy of your calculations. #### Step 6: Practice and Reinforcement The best way to become proficient in unit conversions is through practice. Work on various practice problems that require you to convert between different units. This will help reinforce your understanding and improve your skills. Additionally, always double-check your work to ensure that your conversions are accurate and that you have used the correct conversion factors. Remember, unit conversions are a fundamental skill in chemistry and other sciences, and mastering this skill will greatly benefit your studies and future scientific endeavors. ### FAQs Here are some frequently asked questions about unit conversions in chemistry: #### 1. How do you do conversions in chemistry? To perform conversions in chemistry, follow these steps: 1. Identify the given quantity and its unit. 2. Determine the desired unit for the final answer. 3. Use appropriate conversion factors to set up an equation. 4. Multiply the given quantity by the conversion factor(s). 5. Cancel out units and calculate the result. #### 2. What is the conversion formula in chemistry? There isn't a single conversion formula in chemistry, but rather a method called dimensional analysis. The general approach is: Given quantity × (Conversion factor) = Desired quantity For example: 5 g × (1 mol / 12 g) = 0.417 mol #### 3. What are some common conversions in chemistry? Common conversions in chemistry include: - Grams to moles (and vice versa) - Liters to milliliters - Celsius to Kelvin - Atmospheres to pascals - Molarity to molality #### 4. Do you have to memorize conversions for chemistry? While memorizing some basic conversions can be helpful, it's more important to understand the concept of dimensional analysis and how to use conversion factors. Many conversions can be looked up when needed, but understanding the process is crucial. #### 5. How do you convert units step by step? To convert units step by step: 1. Write down the given quantity with its unit. 2. Identify the target unit. 3. Find the appropriate conversion factor(s). 4. Set up the equation, placing the conversion factor(s) so units cancel out. 5. Multiply the numbers and simplify the units. 6. Calculate the final answer and include the correct unit. ### Prerequisite Topics for Unit Conversions in Chemistry Understanding unit conversions in chemistry is crucial for success in the field, but it's equally important to grasp the foundational concepts that support this skill. One of the most fundamental prerequisites is balancing chemical equations. This skill is essential because it ensures that the number of atoms of each element is conserved during a chemical reaction, which directly impacts the quantities and units involved in conversions. Before diving into complex unit conversions, students should have a solid grasp of chemical reactions. Understanding how substances interact and transform is key to interpreting the units and quantities involved in these processes. This knowledge forms the basis for more advanced calculations and conversions in chemistry. Another critical prerequisite is proficiency in significant figures. In chemistry, precision is paramount, and significant figures help maintain the appropriate level of accuracy in measurements and calculations. When performing unit conversions, knowing how to properly use and report significant figures ensures that the final results are meaningful and scientifically sound. Closely related to significant figures is the concept of scientific notation. This mathematical tool is indispensable in chemistry, where quantities can range from incredibly small to astronomically large. Mastering scientific notation allows students to express and manipulate these values efficiently, making complex unit conversions more manageable. The interplay between these prerequisite topics creates a strong foundation for tackling unit conversions in chemistry. For instance, when converting between moles and grams in a chemical reaction, students must apply their knowledge of balancing equations to determine the correct stoichiometric ratios. They then use scientific notation to handle large or small numbers and apply significant figures rules to ensure the accuracy of their final answer. Moreover, understanding chemical reactions provides context for why certain unit conversions are necessary and how they relate to real-world chemical processes. This background knowledge helps students interpret problem statements and choose the appropriate conversion factors. By mastering these prerequisite topics, students develop a more intuitive understanding of unit conversions in chemistry. They can approach problems with confidence, knowing they have the tools to break down complex conversions into manageable steps. This comprehensive understanding not only aids in solving specific problems but also contributes to a deeper appreciation of chemistry's quantitative aspects. In conclusion, while unit conversions might seem like a standalone topic, they are deeply interconnected with these fundamental concepts. Investing time in solidifying these prerequisites will pay dividends throughout a student's chemistry education and beyond, enabling them to tackle increasingly complex problems with ease and accuracy. In this lesson, we will learn: • The units of measurements commonly used in chemistry. • How to use the unit conversion method and the reason it is valuable. • Practical examples of using the unit conversion method to do calculations in chemistry. Notes: • In any problem where information you have has different units to the information you're being asked for, you'll need to do a unit conversion. • Chemistry calculations involve units like number of moles (units: mol), the mass of a substance (units: g), the volume of a gas, liquid or solution (units: L) and others. • Calculations in chemistry can be solved by breaking down questions into segments: • An unknown quantity to be found - the answer to the question. • An initial quantity to be converted into the units of the unknown quantity. • A conversion factor(s) linking the unknown quantity and the initial quantity. • A conversion factor is an expression as a fraction that equates one unit to another. For example: $\frac{1\;min}{60\;s}$ and $\frac{60\;s}{1\;min}$ • Because the value of both terms in the unit conversion are equal (60 seconds is equal to 1 minute), when multiplying by a unit conversion the value of the expression doesn't change. • This also means you can arrange either term (seconds or mins) on the top or the bottom; arrange it so that your original units cancel and you convert to the new units. This is why it is known as a conversion factor. • CONVERSION FACTORS WILL CHANGE THE UNITS WITHOUT CHANGING THE VALUE! • To solve calculations using the unit conversion method, the following steps should be done in order: • Identify the unknown quantity to be found – this should be written with units and put one side of an equation. • Identify the initial quantity the question has given you – this starts, with units, on the other side of the equation. • Apply the unit conversion(s) by multiplying it with the initial quantity you were given. • This works even if multiple unit conversions are necessary – this method also encourages you to display your working clearly so any mistakes are usually easy to spot! • For example: If there are 6 eggs in a box, how many eggs would be in 4.5 boxes? • For example (part 2): If an egg costs$2 each, how much does 3 dozen eggs cost?
• This method can be used beyond chemistry to solve any problem involving a known quantity that can be converted into another unknown quantity.