What is the Relation between Cp and Cv?

S7-SA4-0038

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Cp and Cv are specific heats of a gas. Cp is the specific heat at constant pressure, meaning it's the heat needed to raise the temperature of a gas by 1 degree Celsius while keeping its pressure constant. Cv is the specific heat at constant volume, which is the heat needed to raise the temperature of a gas by 1 degree Celsius while keeping its volume constant.

Simple Example

Quick Example

Imagine you're heating water for chai. If you heat it in an open pot (constant pressure), some energy goes into expanding the steam. If you heat it in a super strong, closed pressure cooker (constant volume), all the energy goes into just raising the water's temperature. That extra work done in the open pot is why Cp is generally greater than Cv.

Worked Example

Step-by-Step

Let's find the relation between Cp and Cv for an ideal gas using the First Law of Thermodynamics.

Step 1: For a process at constant volume (Cv), the heat absorbed (dQv) is used only to increase the internal energy (dU). So, dQv = dU = Cv dT.

---

Step 2: For a process at constant pressure (Cp), the heat absorbed (dQp) increases internal energy (dU) AND does work (dW) due to expansion. So, dQp = dU + dW.

---

Step 3: We know dW = P dV (work done by gas) and dQp = Cp dT.

---

Step 4: Substitute dQp and dW into the equation from Step 2: Cp dT = dU + P dV.

---

Step 5: From Step 1, we know dU = Cv dT. Substitute this into the equation from Step 4: Cp dT = Cv dT + P dV.

---

Step 6: For an ideal gas, we have the ideal gas equation: PV = nRT. For 1 mole of gas, PV = RT. Differentiating with respect to temperature (at constant pressure), P dV = R dT.

---

Step 7: Substitute P dV = R dT into the equation from Step 5: Cp dT = Cv dT + R dT.

---

Step 8: Divide the entire equation by dT: Cp = Cv + R.

Answer: The relation is Cp - Cv = R, where R is the Universal Gas Constant.

Why It Matters

Understanding Cp and Cv is crucial for engineers designing engines for cars and rockets, as it helps calculate efficiency. In climate science, it helps predict how gases in the atmosphere respond to temperature changes. Knowing this relationship is key for careers in mechanical engineering, aerospace, and environmental science.

Common Mistakes

MISTAKE: Thinking Cp and Cv are always equal. | CORRECTION: Cp is always greater than Cv for gases because at constant pressure, some heat is used for expansion work, not just increasing temperature.

MISTAKE: Forgetting that the relation Cp - Cv = R is specifically for ideal gases. | CORRECTION: While generally true, this exact relation is derived for ideal gases. Real gases have slight deviations.

MISTAKE: Confusing the units of R (Universal Gas Constant). | CORRECTION: R is typically in Joules per mole Kelvin (J/mol.K) or calories per mole Kelvin (cal/mol.K). Ensure all units are consistent in calculations.

Practice Questions

Try It Yourself

QUESTION: If the specific heat at constant volume (Cv) for a gas is 12.5 J/mol.K and the Universal Gas Constant (R) is 8.314 J/mol.K, what is the specific heat at constant pressure (Cp) for this gas? | ANSWER: Cp = Cv + R = 12.5 + 8.314 = 20.814 J/mol.K

QUESTION: For a diatomic gas, the ratio of specific heats (gamma = Cp/Cv) is approximately 1.4. If R = 8.314 J/mol.K, find the values of Cp and Cv. | ANSWER: We have Cp/Cv = 1.4 and Cp - Cv = R. From the first, Cp = 1.4 Cv. Substitute into the second: 1.4 Cv - Cv = R => 0.4 Cv = R => Cv = R / 0.4 = 8.314 / 0.4 = 20.785 J/mol.K. Then Cp = 1.4 * 20.785 = 29.1 J/mol.K.

QUESTION: A gas absorbs 500 J of heat at constant volume, causing its temperature to rise by 10 K. If the same amount of heat is supplied at constant pressure to the same amount of gas, and the work done by the gas is 150 J, what is the change in internal energy in the second case? | ANSWER: At constant volume, heat absorbed (Qv) = change in internal energy (dU). So, dU = 500 J. At constant pressure, heat absorbed (Qp) = dU + work done (W). We are given Qp = 500 J and W = 150 J. So, 500 J = dU + 150 J. Therefore, dU = 500 - 150 = 350 J.

MCQ

Quick Quiz

Which of the following statements is true regarding Cp and Cv for an ideal gas?

Cp is always less than Cv

Cp is always equal to Cv

Cp is always greater than Cv

Cp and Cv are unrelated

The Correct Answer Is:

Cp is always greater than Cv for an ideal gas because at constant pressure, some of the heat energy supplied is used to do work by expanding the gas, in addition to increasing its internal energy.

Real World Connection

In the Real World

This concept is vital in understanding how air conditioners or refrigerators work. The refrigerant gas inside absorbs and releases heat. Engineers use Cp and Cv values to design these systems efficiently, ensuring your home stays cool during hot Indian summers without consuming too much electricity.

Key Vocabulary

Key Terms

SPECIFIC HEAT: The amount of heat required to raise the temperature of a unit mass (or mole) of a substance by one degree Celsius or Kelvin. | CONSTANT PRESSURE: A process where the pressure of the system remains unchanged. | CONSTANT VOLUME: A process where the volume of the system remains unchanged. | UNIVERSAL GAS CONSTANT (R): A physical constant relating energy, temperature, and amount of substance for ideal gases. | INTERNAL ENERGY: The total energy contained within a thermodynamic system.

What's Next

What to Learn Next

Now that you understand Cp and Cv, explore the concept of 'Adiabatic Processes.' These processes involve no heat exchange, and understanding Cp/Cv (gamma) is crucial for analyzing them, which is important for things like rocket propulsion and sound waves!