Animated Solution for Physics - Kinetic Theory: A $15\text{ g}$ mass of nitrogen gas is enclosed in a vessel at a temperature $27^{\circ }\text{C}$. Amount of heat transferred to the gas, so that rms velocity of molecules is doubled is about (Take, $R=8.3\text{ J /K -mole}$)

Trapped Nitrogen Imagine a closed vessel filled with $15\text{ g}$ of nitrogen gas at a comfortable $27^{\circ }\text{C}$. We want to pump heat into this vessel until the molecules are zipping around twice as fast as they were initially.

Before we do any math, we must convert our temperature to the absolute Kelvin scale. This is a non-negotiable rule in thermodynamics! So, our initial temperature is $T_1=27+273=300\text{ K}$.

How is the speed of these molecules related to the temperature of the gas? The kinetic theory of gases tells us that the root mean square velocity ($v_{\text{rms}}$) is directly proportional to the square root of the absolute temperature:

This means $v_{\text{rms}}\propto \sqrt{T}$. If we want to double the speed (multiply by 2), we must multiply the temperature by $2^2$, which is 4.

Therefore, our final temperature must be $T_2=4\times 300\text{ K}=1200\text{ K}$.

The volume is strictly constant. When we add heat at constant volume, all the heat goes directly into increasing the internal energy of the gas. The formula for this is:

Let's gather our ingredients for this equation: 1. Number of moles ($n$): We have $15\text{ g}$ of nitrogen. Since nitrogen exists as diatomic ${\text{N}}_2$ molecules, its molar mass is $28\text{ g/mol}$. So, $n=\frac{15}{28}$. 2. Molar heat capacity ($C_V$): For a diatomic gas like nitrogen at these temperatures, the degree of freedom is 5. Thus, $C_V=\frac{5}{2}R$. 3. Change in temperature ($\Delta T$): $\Delta T=1200\text{ K}-300\text{ K}=900\text{ K}$.

Now, we just plug everything into our heat equation:

Converting this to kilojoules, we get approximately $10\text{ kJ}$. This is the exact amount of thermal energy required to double the microscopic chaos inside the vessel!