A diesel engine takes in 1 mole of air at 300 K, 1 atm pressure and compresses it adiabatically to `(1)/(32)th` of the original volume. Considering air as a diatomic ideal gas, the change in temperature is

Options

A. 900 K
B. 1200 K
C. 600 K
D. 2400 K

Answer

To solve the problem of the change in temperature of air compressed adiabatically in a diesel engine, we can follow these steps:

### Step 1: Understanding the Problem
We are given:
- Initial temperature (T1) = 300 K
- Initial pressure (P1) = 1 atm
- Initial volume (V1) = V (unknown)
- Final volume (V2) = V/32 (compressed to 1/32 of the original volume)
- Number of moles (n) = 1 mole
- Air is treated as a diatomic ideal gas, so we use γ (gamma) = 7/5.

### Step 2: Use the Adiabatic Process Equation
For an adiabatic process, we can use the relation:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
Where:
- T1 = Initial temperature
- T2 = Final temperature
- V1 = Initial volume
- V2 = Final volume
- γ = Heat capacity ratio (7/5 for diatomic gases)

### Step 3: Substitute the Known Values
We know:
- V2 = V1/32
- Therefore, we can substitute V2 in the equation:
\[ T_1 V_1^{\gamma - 1} = T_2 (V_1/32)^{\gamma - 1} \]

### Step 4: Simplifying the Equation
Rearranging the equation gives us:
\[ T_2 = T_1 \left( \frac{V_1}{V_1/32} \right)^{\gamma - 1} \]
\[ T_2 = T_1 \left( 32 \right)^{\gamma - 1} \]

### Step 5: Calculate γ - 1
Now we calculate:
\[ \gamma - 1 = \frac{7}{5} - 1 = \frac{2}{5} \]

### Step 6: Substitute γ - 1 into the Equation
Now substitute γ - 1 back into the equation for T2:
\[ T_2 = 300 \left( 32 \right)^{\frac{2}{5}} \]

### Step 7: Calculate \( 32^{\frac{2}{5}} \)
We can express 32 as \( 2^5 \):
\[ 32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2 = 4 \]

### Step 8: Final Calculation of T2
Now substitute back to find T2:
\[ T_2 = 300 \times 4 = 1200 \, K \]

### Step 9: Change in Temperature
The change in temperature (ΔT) can be calculated as:
\[ \Delta T = T_2 - T_1 = 1200 \, K - 300 \, K = 900 \, K \]

### Final Answer
The change in temperature is \( 900 \, K \).

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