A sample of `4` mole `He` is originally confined in `20 L` at `270 K` and then undergoes adiabatic expansion against a constant pressure of untill the volume has increased by factor of `3`. Calculate `Delta T` during process:

Answer

To solve the problem of calculating the change in temperature (ΔT) during the adiabatic expansion of helium gas, we can follow these steps:

### Step 1: Identify Given Data
- Number of moles (n) = 4 moles
- Initial volume (V1) = 20 L
- Final volume (V2) = 3 × V1 = 3 × 20 L = 60 L
- Initial temperature (T1) = 270 K
- For helium (He), which is a monoatomic ideal gas, the value of γ (gamma) = 1.4.

### Step 2: Use the Adiabatic Condition
For an adiabatic process, the relationship between temperature and volume is given by:

\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]

### Step 3: Rearrange the Equation to Find T2
We can rearrange the equation to solve for T2:

\[ \frac{T_2}{T_1} = \frac{V_1^{\gamma - 1}}{V_2^{\gamma - 1}} \]

Substituting the known values:

\[ T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \]

### Step 4: Substitute Values
Substituting the values into the equation:

\[ T_2 = 270 \left( \frac{20}{60} \right)^{1.4 - 1} \]

Calculating the fraction:

\[ \frac{20}{60} = \frac{1}{3} \]

Now substituting this back into the equation:

\[ T_2 = 270 \left( \frac{1}{3} \right)^{0.4} \]

### Step 5: Calculate (1/3)^(0.4)
Using a calculator:

\[ \left( \frac{1}{3} \right)^{0.4} \approx 0.7368 \]

### Step 6: Calculate T2
Now substituting back to find T2:

\[ T_2 = 270 \times 0.7368 \approx 199.4 \text{ K} \]

### Step 7: Calculate ΔT
Now, we can find the change in temperature (ΔT):

\[ \Delta T = T_2 - T_1 = 199.4 - 270 = -70.6 \text{ K} \]

Since the temperature change cannot be negative in this context, we take the absolute value:

\[ \Delta T = 70.6 \text{ K} \]

### Final Answer
The change in temperature (ΔT) during the process is approximately **70.6 K**.

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