X and Y are two equal size containers. X contains a gas A at a temperature `75^@ `C and Y contains a gas B at a temperature `40^@ `C. Each gas behaves as an ideal gas and specific heat at constant pressure for both gases has the same value.
It can be concluded that the

Options

A. internal energy of a gas in container X is more than that of the gas in container Y.
B. internal energy of a gas in container X is less than that of the gas in container Y.
C. internal energy of a gas in container X could be more or could also be less than that of the gas in container Y.
D. internal energy of a gas in container X has the same value as that of the gas in container.

Answer

To solve the problem, we need to analyze the information given about the two gases in containers X and Y.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Container X contains gas A at a temperature of \(75^\circ C\).
- Container Y contains gas B at a temperature of \(40^\circ C\).
- Both gases behave as ideal gases.
- The specific heat at constant pressure (\(C_P\)) for both gases is the same.

2. **Understanding Internal Energy**:
- The internal energy (\(U\)) of an ideal gas can be expressed in terms of the number of moles (\(n\)), the specific heat at constant volume (\(C_V\)), and the change in temperature (\(\Delta T\)):
\[ \Delta U = n C_V \Delta T \]
- Since \(C_P\) is the same for both gases, we can infer that \(C_V\) will also be the same for both gases, given that \(C_P - C_V = R\) (where \(R\) is the universal gas constant).

3. **Calculating Temperature Changes**:
- The temperature of gas A is \(75^\circ C\) and that of gas B is \(40^\circ C\).
- The change in temperature for gas A compared to a reference point (let's say \(0^\circ C\)) is \(75\) degrees.
- The change in temperature for gas B is \(40\) degrees.
- Therefore, we can say:
\[ \Delta T_A = 75 - 0 = 75 \quad \text{and} \quad \Delta T_B = 40 - 0 = 40 \]

4. **Comparing Internal Energies**:
- Since both gases have the same \(C_V\) and the same number of moles (assuming equal volumes and ideal gas behavior), the internal energy changes can be compared:
\[ \Delta U_A = n C_V \Delta T_A = n C_V (75) \]
\[ \Delta U_B = n C_V \Delta T_B = n C_V (40) \]
- From this, we can see that the internal energy change for gas A is greater than that for gas B because \(75 > 40\).

5. **Conclusion**:
- However, we cannot definitively conclude that the internal energy of gas A is greater than that of gas B without knowing the number of moles or the molecular weight of each gas. The internal energy could be greater, less, or equal depending on the molecular weights of gases A and B.
- Therefore, the conclusion is that the internal energy of gas A may be greater than, less than, or equal to the internal energy of gas B.

### Final Answer:
The internal energy of gas A may be greater than, less than, or equal to the internal energy of gas B.

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