An ideal diatomic gas is taken through an adiabatic process in which density increases to 32 times. If pressure increases to 'n' times. Find n

Options

A. 4
B. 8
C. 64
D. 128

Answer

To solve the problem, we need to analyze the adiabatic process of an ideal diatomic gas and how the changes in density affect the pressure. Here’s a step-by-step solution:

### Step 1: Understand the relationship between density, mass, and volume
The density (ρ) of a gas is defined as:
\[ \rho = \frac{m}{V} \]
where \(m\) is the mass and \(V\) is the volume. Since the mass of the gas remains constant during the process, if the density increases to 32 times its initial value, we can express this as:
\[ \rho' = 32 \rho \]

### Step 2: Relate the change in density to the change in volume
From the definition of density, we can write:
\[ \rho' = \frac{m}{V'} \quad \text{and} \quad \rho = \frac{m}{V} \]
Thus, we have:
\[ \frac{\rho'}{\rho} = \frac{V}{V'} = 32 \]
This implies:
\[ \frac{V'}{V} = \frac{1}{32} \]

### Step 3: Use the adiabatic condition for an ideal gas
For an adiabatic process involving an ideal diatomic gas, the relationship between pressure (P), volume (V), and the adiabatic exponent (γ) is given by:
\[ PV^{\gamma} = \text{constant} \]
For a diatomic gas, the adiabatic exponent \(γ\) is:
\[ \gamma = \frac{7}{5} \]

### Step 4: Set up the equation for the initial and final states
Let \(P\) be the initial pressure and \(P'\) be the final pressure. The relationship can be expressed as:
\[ P' V'^{\frac{7}{5}} = P V^{\frac{7}{5}} \]
Dividing both sides by \(P\):
\[ \frac{P'}{P} = \left(\frac{V}{V'}\right)^{\frac{7}{5}} \]

### Step 5: Substitute the volume ratio
From Step 2, we know:
\[ \frac{V}{V'} = 32 \]
Substituting this into the equation gives:
\[ \frac{P'}{P} = 32^{\frac{7}{5}} \]

### Step 6: Calculate \(32^{\frac{7}{5}}\)
We can express \(32\) as \(2^5\):
\[ 32^{\frac{7}{5}} = (2^5)^{\frac{7}{5}} = 2^7 \]
Thus:
\[ \frac{P'}{P} = 2^7 \]

### Step 7: Find the value of \(n\)
From the equation above, we see that:
\[ P' = 2^7 P \]
This means that the pressure increases by a factor of \(n = 2^7 = 128\).

### Final Answer
Thus, the value of \(n\) is:
\[ \boxed{128} \]