Question 892: Thermodynamics & KTG

Question

Which one of the following is NOT a correct expression for an ideal gas?
[$$\mathrm{C_p}=$$ Molar specific heat of a gas at constant pressure,
$$\mathrm{C_v}=$$ Molar specific heat of a gas at constant volume,
$$\mathrm{Y}=$$ Ratio of two specific heats of a gas,
$$\mathrm{R}=$$ Universal gas constant]

Accepted Answer

To determine which of the expressions given for an ideal gas is NOT correct, we need to recall some fundamental relationships between the specific heats and the gas constant for an ideal gas.
For an ideal gas, the following relationships hold:
1. $$\mathrm{C_p} - \mathrm{C_v} = \mathrm{R}$$
2. $$\gamma = \frac{\mathrm{C_p}}{\mathrm{C_v}}$$
3. $$\mathrm{R} = \mathrm{C_v}(\gamma - 1)$$
4. $$\mathrm{C_v} = \frac{\mathrm{R}}{\gamma - 1}$$
Let's analyze each option:
Option A: $$\mathrm{C_v} = \mathrm{C_p} + \mathrm{R}$$
This is incorrect because according to the first relationship, $$\mathrm{R} = \mathrm{C_p} - \mathrm{C_v}$$, and therefore, $$\mathrm{C_v}$$ cannot be greater than $$\mathrm{C_p}$$.
Option B: $$\mathrm{R} = \mathrm{C_v}(\gamma - 1)$$
This is correct as it matches the third relationship.
Option C: $$\frac{\mathrm{C_v}}{\mathrm{C_p}} = \frac{1}{\gamma}$$
This is correct since $$\gamma = \frac{\mathrm{C_p}}{\mathrm{C_v}}$$ therefore, $$\frac{\mathrm{C_v}}{\mathrm{C_p}} = \frac{1}{\gamma}$$.
Option D: $$R = \frac{\mathrm{C_p}(\gamma - 1)}{\gamma}$$
This can be derived from the given relationships. By substituting $$\mathrm{C_v} = \frac{\mathrm{C_p}}{\gamma}$$ into $$\mathrm{R} = \mathrm{C_v}(\gamma - 1)$$, we get $$R = \frac{\mathrm{C_p}}{\gamma} (\gamma - 1) = \frac{\mathrm{C_p}(\gamma - 1)}{\gamma}$$ so this is correct.
Therefore, the incorrect expression is:
Option A: $$C_v=C_p+R$$

Answer

$$C_v=C_p+R$$

Answer

$$R=C_v(\gamma-1)$$

Answer

$$\frac{C_v}{\mathrm{C}_{\mathrm{p}}}=\frac{1}{\gamma}$$

Answer

$$R=\frac{C_{\mathrm{p}}(\gamma-1)}{\gamma}$$

Thermodynamics & KTG

Which one of the following is NOT a correct expression for an ideal gas? [$\mathrm{C_p}= Molar specific heat of a gas at constant pressure, \mathrm{C_v}= Molar specific heat of a gas at constant volume, \mathrm{Y}= Ratio of two specific heats of a gas, \mathrm{R}=$ Universal gas constant]

Choose the correct answer: