An ideal diatomic gas is expanded so that the amount of heat transferred to the gas is equal to the decrease in its internal energy.
The process can be represented by the equation `TV^(n) =` constant, where the value of `n` is

Options

A. `n=(7)/(5)`
B. `n=(1)/(5)`
C. `n=(3)/(2)`
D. `n=(3)/(5)`

Explanation

To solve the problem, we need to analyze the situation where an ideal diatomic gas is expanded such that the heat transferred to the gas equals the decrease in its internal energy. We will derive the value of \( n \) in the equation \( TV^n = \text{constant} \).

### Step 1: Understand the relationship between heat transfer and internal energy
Given that the heat transferred \( dQ \) is equal to the negative change in internal energy \( -dU \), we can write:
\[ dQ = -dU \]

### Step 2: Use the first law of thermodynamics
From the first law of thermodynamics, we know:
\[ dQ = dU + dW \]
Substituting \( dQ = -dU \) into this equation gives:
\[ -dU = dU + dW \]
This implies:
\[ dW = -2dU \]

### Step 3: Relate \( dQ \) and \( dU \) for an ideal gas
For an ideal diatomic gas, the change in internal energy \( dU \) is given by:
\[ dU = n C_v dT \]
where \( C_v = \frac{5}{2} R \) for a diatomic gas. Therefore:
\[ dU = n \left(\frac{5}{2} R\right) dT \]

### Step 4: Substitute \( dU \) into the expression for \( dW \)
Now substituting \( dU \) into the expression for \( dW \):
\[ dW = -2 \left(n \frac{5}{2} R dT\right) = -5nR dT \]

### Step 5: Relate \( dW \) to \( P \) and \( dV \)
Using the ideal gas law, \( PV = nRT \), we can express \( dW \) as:
\[ dW = P dV = \frac{nRT}{V} dV \]
Equating the two expressions for \( dW \):
\[ -5nR dT = \frac{nRT}{V} dV \]

### Step 6: Simplify the equation
Dividing both sides by \( nR \) (assuming \( n \neq 0 \)):
\[ -5 dT = \frac{T}{V} dV \]
Rearranging gives:
\[ \frac{dV}{V} = -5 \frac{dT}{T} \]

### Step 7: Integrate both sides
Integrating both sides:
\[ \int \frac{dV}{V} = -5 \int \frac{dT}{T} \]
This results in:
\[ \ln V = -5 \ln T + \text{constant} \]
Exponentiating both sides gives:
\[ V = C T^{-5} \]
where \( C \) is a constant.

### Step 8: Rearranging to find the equation
Rearranging this equation, we have:
\[ TV^5 = \text{constant} \]
Thus, we can express it in the form \( TV^n = \text{constant} \) where \( n = -5 \).

### Final Answer
The value of \( n \) is:
\[ n = \frac{1}{5} \]