Question

If kinetic energy of one mole of monatomic gas is 24.4J, then temperature is(R = 8.314)

Hint:

Remember the degrees of freedom ($f$):
- Monatomic gas: $f = 3$ (only translational). $E = \frac{3}{2}nRT$
- Diatomic gas: $f = 5$ (3 translational + 2 rotational at normal temps). $E = \frac{5}{2}nRT$

Answer 1

Step 1: Understanding the Concept:
According to the kinetic theory of gases, the translational kinetic energy of an ideal gas is directly proportional to its absolute temperature. A monatomic gas (like Helium or Neon) has only 3 translational degrees of freedom.

Step 2: Key Formula or Approach:
The total translational kinetic energy ($K$) for $n$ moles of a monatomic ideal gas is given by:
\[ K = \frac{3}{2} n R T \]
Where:
$n$ = number of moles
$R$ = universal gas constant
$T$ = absolute temperature in Kelvin

Step 3: Detailed Explanation:
Given values:
Kinetic energy, $K = 24.4$ J
Number of moles, $n = 1$ mole
Gas constant, $R = 8.314$ J/(mol$\cdot$K)
Rearrange the formula to solve for temperature $T$:
\[ T = \frac{2K}{3nR} \]
Substitute the known values:
\[ T = \frac{2 \times 24.4}{3 \times 1 \times 8.314} \]
\[ T = \frac{48.8}{24.942} \]
\[ T \approx 1.9565 \text{ K} \]
Rounding to a reasonable number of significant figures gives approximately 1.96 K.

Step 4: Final Answer:
The temperature is approximately $1.96$ K.