An ideal gas occupies a volume of $2\,m^3$ at a pressure of $3 \times 10^6\, Pa$. The energy of the gas is:

Question

In a mixture of gases, the average number of degrees of freedom per molecule is 6. If the rms speed of the molecule is $c$, what is the velocity of sound in the gas?

Answer 1

To find the energy of an ideal gas, we must relate the given information to the internal energy formula. For an ideal gas, the internal energy (U) is equated to:

U = \frac{3}{2} nRT

where:

n is the number of moles
R is the universal gas constant
T is the temperature in Kelvin

However, we are given the volume and pressure, therefore, we must use the ideal gas equation:

PV = nRT

We can express the number of moles n in terms of P, V, and T as: n = \frac{PV}{RT}

Substitute the expression for n into the formula for the internal energy:

U = \frac{3}{2} \frac{PV}{RT} \cdot RT = \frac{3}{2} PV

Substituting the provided values into the above equation:

Volume, V = 2\, m^3
Pressure, P = 3 \times 10^6\, Pa

Thus,

U = \frac{3}{2} \times 3 \times 10^6 \times 2 = 9 \times 10^6\, J

Therefore, the energy of the gas is indeed 9 \times 10^6\, J.

The correct answer chosen from the options is:

9 \times 10^{-6} J which is incorrect based on the energy calculated. Rechecking the solution seems that provided calculation aligns with given data.

Answer 2