The temperature of a sample of a gas is raised from `127^(@)C` to `527^(@)C` .The average kinetic energy of the gas

Options

A. does not change
B. is doubled
C. is halved
D. cannot be calculated

Answer

To solve the problem of how the average kinetic energy of a gas changes when its temperature is raised from \(127^\circ C\) to \(527^\circ C\), we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and temperature
The average kinetic energy (KE) of a gas is directly proportional to its absolute temperature (in Kelvin). The formula for average kinetic energy is given by:
\[ KE = \frac{3}{2} k T \]
where \(k\) is the Boltzmann constant and \(T\) is the temperature in Kelvin.

### Step 2: Convert the temperatures from Celsius to Kelvin
To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]

- For the initial temperature \(T_1 = 127^\circ C\):
\[ T_1 = 127 + 273 = 400 \, K \]

- For the final temperature \(T_2 = 527^\circ C\):
\[ T_2 = 527 + 273 = 800 \, K \]

### Step 3: Determine the ratio of the average kinetic energies
Since the average kinetic energy is directly proportional to the temperature, we can set up the following ratio:
\[ \frac{KE_1}{KE_2} = \frac{T_1}{T_2} \]
Substituting the values we found:
\[ \frac{KE_1}{KE_2} = \frac{400}{800} = \frac{1}{2} \]

### Step 4: Solve for the final kinetic energy
From the ratio, we can express the final kinetic energy in terms of the initial kinetic energy:
\[ KE_2 = 2 \times KE_1 \]
This means that the average kinetic energy doubles when the temperature is raised from \(127^\circ C\) to \(527^\circ C\).

### Conclusion
Thus, the average kinetic energy of the gas at \(527^\circ C\) is double that at \(127^\circ C\).