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Find Cas/aT) in terms of measurable. Properties CP-V-T)

Find Cas/aT) in terms of measurable. Properties CP-V-T)

Chemistry
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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**Title: How to Determine Cᵥ (Heat Capacity at Constant Volume) Using Measurable Properties (P, V, and T)** --- **Introduction:** In thermodynamics, it's often necessary to determine specific heat capacities. One such heat capacity is \( C_V \), which is the heat capacity at constant volume. This guide provides a step-by-step process to find \( C_V \) using measurable properties: Pressure (P), Volume (V), and Temperature (T). --- **Detailed Instruction:** 1. **Understand the Fundamental Concepts:** * **Pressure (P):** The force exerted per unit area. * **Volume (V):** The amount of space occupied by a substance. * **Temperature (T):** A measure of the thermal energy within a system. 2. **Apply the Ideal Gas Law:** The relationship between P, V, and T for an ideal gas is described by the equation: \[ PV = nRT \] where \( n \) is the number of moles of the gas and \( R \) is the universal gas constant. 3. **Relate Heat Capacity to the Gas Properties:** The heat capacity at constant volume (\( C_V \)) relates the change in heat added to the change in temperature, without consideration of volume change. Mathematically, it's: \[ C_V = \left( \frac{\partial Q}{\partial T} \right)_V \] 4. **Use the Thermodynamic Identity:** For an ideal gas, the specific heat at constant volume can also be related using \[ C_V = \frac{3}{2}nR \text{ (for a monoatomic ideal gas)} \] 5. **Measurement and Calculation:** * Measure the initial and final temperatures (T1 and T2). * Determine the heat added (Q) to the system while keeping the volume constant. * Calculate \( C_V \) using: \[ C_V = \frac{Q}{n(T_2 - T_1)} \] --- **Conclusion:** By understanding and applying the fundamental properties and relationships between pressure, volume, and temperature, one can effectively determine the heat capacity at constant volume (\( C_V \)). Practical laboratory measurements of these properties will allow you to calculate \( C_V \) for various gases under ideal
Transcribed Image Text:**Title: How to Determine Cᵥ (Heat Capacity at Constant Volume) Using Measurable Properties (P, V, and T)** --- **Introduction:** In thermodynamics, it's often necessary to determine specific heat capacities. One such heat capacity is \( C_V \), which is the heat capacity at constant volume. This guide provides a step-by-step process to find \( C_V \) using measurable properties: Pressure (P), Volume (V), and Temperature (T). --- **Detailed Instruction:** 1. **Understand the Fundamental Concepts:** * **Pressure (P):** The force exerted per unit area. * **Volume (V):** The amount of space occupied by a substance. * **Temperature (T):** A measure of the thermal energy within a system. 2. **Apply the Ideal Gas Law:** The relationship between P, V, and T for an ideal gas is described by the equation: \[ PV = nRT \] where \( n \) is the number of moles of the gas and \( R \) is the universal gas constant. 3. **Relate Heat Capacity to the Gas Properties:** The heat capacity at constant volume (\( C_V \)) relates the change in heat added to the change in temperature, without consideration of volume change. Mathematically, it's: \[ C_V = \left( \frac{\partial Q}{\partial T} \right)_V \] 4. **Use the Thermodynamic Identity:** For an ideal gas, the specific heat at constant volume can also be related using \[ C_V = \frac{3}{2}nR \text{ (for a monoatomic ideal gas)} \] 5. **Measurement and Calculation:** * Measure the initial and final temperatures (T1 and T2). * Determine the heat added (Q) to the system while keeping the volume constant. * Calculate \( C_V \) using: \[ C_V = \frac{Q}{n(T_2 - T_1)} \] --- **Conclusion:** By understanding and applying the fundamental properties and relationships between pressure, volume, and temperature, one can effectively determine the heat capacity at constant volume (\( C_V \)). Practical laboratory measurements of these properties will allow you to calculate \( C_V \) for various gases under ideal
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