For 1 mole of ideal gas which of the following statements must be true
a) U & H depend only on temperature
b) Compressibility factor (Z) can not be 1.
c) `C_p-C_v` =R
d) `triangleU = C_vdT` for all processes

Options

A. a,c,d
B. b,c,d,
C. c,d
D. a,c

Answer

To solve the question regarding the properties of 1 mole of an ideal gas, let's analyze each statement step by step:

### Step 1: Evaluate Statement A
**Statement A**: U and H depend only on temperature.

- **Internal Energy (U)**: For an ideal gas, the internal energy is a function of temperature only. This is because the internal energy is related to the kinetic energy of the gas molecules, which depends solely on temperature.

- **Enthalpy (H)**: Similarly, enthalpy is also a function of temperature for an ideal gas. The relationship can be expressed as:
\[ H = U + PV \]
For an ideal gas, since \(PV = nRT\), substituting this into the enthalpy equation shows that H also depends only on temperature.

**Conclusion**: Statement A is **true**.

### Step 2: Evaluate Statement B
**Statement B**: Compressibility factor (Z) cannot be 1.

- **Compressibility Factor (Z)**: Defined as \(Z = \frac{PV}{RT}\). For an ideal gas, the equation of state is \(PV = nRT\). If we substitute \(n = 1\) mole, we find:
\[ Z = \frac{PV}{RT} = \frac{RT}{RT} = 1 \]

**Conclusion**: Statement B is **false** because the compressibility factor can indeed be 1 for an ideal gas.

### Step 3: Evaluate Statement C
**Statement C**: \(C_p - C_v = R\).

- This is a well-known relation for ideal gases. The difference between the heat capacities at constant pressure and constant volume for 1 mole of an ideal gas is given by:
\[ C_p - C_v = R \]

**Conclusion**: Statement C is **true**.

### Step 4: Evaluate Statement D
**Statement D**: \(\Delta U = C_v dT\) for all processes.

- The equation \(\Delta U = nC_v dT\) holds true for processes where the heat capacity at constant volume is applicable. However, this relation is not valid for all processes, especially for isothermal processes where \(\Delta U = 0\) (since temperature does not change).

**Conclusion**: Statement D is **false** because \(\Delta U = C_v dT\) is not applicable for all types of processes.

### Final Conclusion
Based on the evaluations:
- Statement A: True
- Statement B: False
- Statement C: True
- Statement D: False

Thus, the correct answer is **A and C are true**.

### Answer: A and C

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