Question

For one mole of an ideal gas, which of these statements must be true?

1) (a), (c) and (d)

2) (a) and (c)

3) (c) and (d)

4) (b), (c) and (d)

• A. $\mathrm{U}$ and $\mathrm{H}$ each depend only on the temperature
• B. Compressibility factor $\mathrm{z}$ is not equal to $1$
• C. ${\mathrm{C}}_{\mathrm{P},\mathrm{m}}-{\mathrm{C}}_{\mathrm{V},\mathrm{m}}=\mathrm{R}$
• D. $\mathrm{dU}={\mathrm{C}}_{\mathrm{v}}\mathrm{dT}$ for any process

Answer 1

Answer: (A), (C), (D)

$\mathrm{dU}={\mathrm{C}}_{\mathrm{v}}\mathrm{dT}$ for any process

The above question can be explained as:

Some relation of ${\mathrm{C}}_{\mathrm{p}}$ and ${\mathrm{C}}_{\mathrm{v}}$ for an ideal gas:

Step 1:

Internal energy($\mathrm{U}$) of an ideal gas only on temperature($\mathrm{T}$), neither on volume($\mathrm{V}$) nor the pressure($\mathrm{P}$)

Step 2:

The heat absorbed by gas is equal to its internal heat engine, when ideal gas undergoes process involving only $\mathrm{P}-\mathrm{V}$work.

Step 3:

Therefore, for an ideal gas,

$$\begin{gathered} \mathrm{Cv}={\left(\mathrm{\delta q}/\mathrm{\delta t}\right)}_{\mathrm{v}} \\ {\mathrm{C}}_{\mathrm{v}}=\mathrm{dV}/\mathrm{dT} \\ \mathrm{dU}={\mathrm{C}}_{\mathrm{v}}\mathrm{dT} \end{gathered}$$

Hence, option D is correct.