Ten moles of a diatomic perfect gas are allowed to expand at constant pressure. The initial volume and temperature are `V_0` and `T_0` respectively. If `7/2RT_0` heat is transferred to the gas, then the final volume and temperature are

Options

A. (a) `1.1 V_0, 1.1 T_0`
B. (b) `0.9V_0, 0.9T_0`
C. (c) `1.1V_0,10/11T_0`
D. (d) `0.9V_0,10/9T_0`

Explanation

To solve the problem step by step, we will use the principles of thermodynamics related to ideal gases, particularly focusing on the heat transfer during an isobaric (constant pressure) process.

### Step 1: Understand the Given Data
We have:
- Number of moles, \( n = 10 \)
- Initial volume, \( V_0 \)
- Initial temperature, \( T_0 \)
- Heat transferred, \( Q = \frac{7}{2} R T_0 \)

### Step 2: Use the Formula for Heat Transfer
For an ideal gas undergoing an isobaric process, the heat transferred can be expressed as:
\[ Q = n C_p \Delta T \]
Where:
- \( C_p \) for a diatomic gas is \( \frac{7}{2} R \)
- \( \Delta T = T_f - T_0 \) (where \( T_f \) is the final temperature)

### Step 3: Substitute the Values into the Heat Equation
Substituting the values into the heat equation:
\[ \frac{7}{2} R T_0 = n C_p (T_f - T_0) \]
\[ \frac{7}{2} R T_0 = 10 \left(\frac{7}{2} R\right) (T_f - T_0) \]

### Step 4: Simplify the Equation
We can cancel \( \frac{7}{2} R \) from both sides:
\[ T_0 = 10 (T_f - T_0) \]

### Step 5: Solve for Final Temperature \( T_f \)
Rearranging the equation:
\[ T_0 = 10 T_f - 10 T_0 \]
\[ 11 T_0 = 10 T_f \]
\[ T_f = \frac{11}{10} T_0 = 1.1 T_0 \]

### Step 6: Use the Ideal Gas Law to Find Final Volume
Since the process is at constant pressure, we can use the ideal gas law, which states that:
\[ \frac{V}{T} = \text{constant} \]
Thus, we have:
\[ \frac{V_f}{T_f} = \frac{V_0}{T_0} \]

### Step 7: Solve for Final Volume \( V_f \)
Substituting \( T_f \):
\[ \frac{V_f}{1.1 T_0} = \frac{V_0}{T_0} \]
This simplifies to:
\[ V_f = 1.1 V_0 \]

### Final Results
Thus, the final volume and temperature of the gas are:
- Final Temperature, \( T_f = 1.1 T_0 \)
- Final Volume, \( V_f = 1.1 V_0 \)