The change in internal energy , when a gas is cooled from `927^(@) " to " 27^(@) C `

Options

A. 1
B. 2
C. 0.75
D. 4

Explanation

To find the change in internal energy when a gas is cooled from 927°C to 27°C, we can follow these steps:

### Step 1: Convert Celsius to Kelvin
First, we need to convert the temperatures from Celsius to Kelvin since the internal energy is proportional to the absolute temperature.

- **Initial Temperature (T1)**:
\[ T1 = 927°C + 273 = 1200 \, K \]

- **Final Temperature (T2)**:
\[ T2 = 27°C + 273 = 300 \, K \]

### Step 2: Use the formula for internal energy
The internal energy (U) of an ideal gas is given by the formula:
\[ U = \frac{3}{2}RT \]
where R is the universal gas constant and T is the absolute temperature in Kelvin.

### Step 3: Calculate the change in internal energy
The change in internal energy (ΔU) can be calculated using the difference in internal energy at the two temperatures:
\[ \Delta U = U2 - U1 = \frac{3}{2}R(T2 - T1) \]

### Step 4: Substitute the values
Now, substituting the values of T1 and T2:
\[ \Delta U = \frac{3}{2}R(300 - 1200) = \frac{3}{2}R(-900) \]

### Step 5: Simplify the expression
Since we are interested in the magnitude of the change:
\[ \Delta U = \frac{3}{2}R \cdot (-900) = -1350R \]
Taking the absolute value, we have:
\[ |\Delta U| = 1350R \]

### Step 6: Calculate the percentage change
To find the percentage change in internal energy, we can express it as:
\[ \text{Percentage Change} = \frac{|\Delta U|}{U1} \times 100 \]

Where \( U1 = \frac{3}{2}R \cdot 1200 \):
\[ U1 = 1800R \]

Thus,
\[ \text{Percentage Change} = \frac{1350R}{1800R} \times 100 = \frac{1350}{1800} \times 100 = 75\% \]

### Conclusion
The change in internal energy when the gas is cooled from 927°C to 27°C is a decrease of 75%.

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