Question 706: Thermodynamics & KTG

Question

The average force applied on the walls of a closed container depends as $\mathrm{T}^{\mathrm{x}}$ where T is the temperature of an ideal gas. The value of $x$ is

Accepted Answer

Let us solve the problem step by step.
Pressure of an Ideal Gas:
According to the kinetic theory of gases, the pressure ($P$) exerted by an ideal gas is given by:
$$    P = \frac{1}{3} \frac{N m \langle v^2 \rangle}{V}    $$
where
$N$ = number of molecules,
$m$ = mass of one molecule,
$\langle v^2 \rangle$ = mean square speed,
$V$ = volume.
Kinetic Energy and Temperature:
The average kinetic energy of one molecule is
$$    \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_B T    $$
Thus,
$$    m \langle v^2 \rangle = 3 k_B T    $$
Expressing Pressure in Terms of Temperature:
Substitute the value from step 2 into the pressure equation:
$$    P = \frac{1}{3} \cdot \frac{N}{V} \cdot (3 k_B T)    $$
$$    P = \frac{N}{V} k_B T    $$
For a fixed number of molecules and volume, $N$ and $V$ are constant.
Force on the Walls:
Pressure ($P$) is force ($F$) per unit area ($A$). So,
$$    P = \frac{F}{A} \implies F = P \cdot A    $$
For a given container (fixed area), $F \propto P$.
Dependence on Temperature:
From above, $P \propto T$
So, $F \propto T$
Value of $x$:
Force on walls $\propto T^x$,
So, $x = 1$.
Final Answer:
Option A
$\boxed{1}$

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Thermodynamics & KTG

The average force applied on the walls of a closed container depends as \mathrm{T}^{\mathrm{x}} where T is the temperature of an ideal gas. The value of x is

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