Question

The temperature of a \(5.00-\mathrm{L}\) container of \(\mathrm{N}_{2}\) gas is increased from \(20^{\circ} \mathrm{C}\) to \(250^{\circ} \mathrm{C}\). If the volume is held constant, predict qualitatively how this change affects the following: (a) the average kinetic energy of the molecules; (b) the rootmean- square speed of the molecules; (c) the strength of the impact of an average molecule with the container walls; \(\mathbf{d}\) ) the total number of collisions of molecules with walls per second.

Short answer

By increasing the temperature of the N2 gas from 20°C to 250°C, the average kinetic energy, root-mean-square speed, strength of impact, and total number of collisions per second all increase.

Step-by-step solution

Identify the relationship between variables

According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the absolute temperature. The root-mean-square speed of the molecules also has a direct relationship with the average kinetic energy. The strength of impact and total number of collisions are related to the molecules' kinetic energy and speed. We can start by converting the given temperatures to Kelvin, and then analyze the effect of increasing temperature on the various parameters.

Convert the given temperatures to Kelvin

The given temperatures are in Celsius. We'll convert them to Kelvin by using the following formula: Absolute Temperature (K) = Temperature (°C) + 273.15 Initial temperature (T1): \(20 + 273.15 = 293.15\,\text{K}\) Final temperature (T2): \(250 + 273.15 = 523.15\,\text{K}\)

Predict the changes#a) Average kinetic energy of the molecules

As we know, the average kinetic energy is directly proportional to the absolute temperature. Since the final temperature is greater than the initial temperature, the average kinetic energy will increase. $$\frac{K.E._{final}}{K.E._{initial}} = \frac{T2}{T1}$$

b) Root-mean-square speed of the molecules

The root-mean-square speed of molecules depends on the average kinetic energy, which we already know increases with an increase in temperature. Therefore, the root-mean-square speed will also increase.

c) Strength of the impact with the container walls

The molecule's strength of impact depends on its kinetic energy and speed. As both of these parameters increase with increasing temperature, the strength of impact will also increase.

d) Total number of collisions of molecules with walls per second

As the molecules have a higher kinetic energy and speed, they will collide with the container walls more frequently. As a result, the total number of collisions of molecules with the walls per second will also increase. In conclusion, by increasing the temperature of the N2 gas from 20°C to 250°C, the average kinetic energy, root-mean-square speed, strength of impact, and total number of collisions per second all increase.

Key concepts

Average Kinetic Energy

The average kinetic energy of a gas is directly related to its temperature. When you heat a gas, you're essentially giving its molecules more energy. Imagine these molecules as tiny balls bouncing around.
The faster they move, the more energy they have.
In the kinetic theory of gases, this energy is calculated as:

• Average kinetic energy per molecule = \(\frac{3}{2}kT\)

where \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature in Kelvin.So, when the temperature of the nitrogen gas in the exercise increases from \(293.15\,\text{K}\) to \(523.15\,\text{K}\), the average kinetic energy of all the gas molecules increases. It means every molecule, on average, is moving energetically than before.

Root-Mean-Square Speed

The root-mean-square speed (RMS speed) is a measure of how fast molecules in a gas are moving on average. It's a bit like taking the average speed of all these bouncy balls we mentioned earlier.
But instead of just averaging, we consider their speed squared, take an average, and then the square root of that number.
The formula for RMS speed is:

• \(v_{rms} = \sqrt{\frac{3RT}{M}}\)

where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.As the temperature of the gas increases, so does the RMS speed. So, when the temperature in our scenario jumps, these tiny balls move even faster, hence the RMS speed increases too.

Gas Temperature

The temperature of a gas is a key player in the kinetic theory. It's not just a measurement of heat but it tells us how energetic the molecules in the gas are.
In the formula for average kinetic energy and RMS speed, absolute temperature (measured in Kelvin) is the variable the others are directly dependent on.
When you convert from Celsius to Kelvin for calculations, for instance, you make sure that zero temperature truly represents zero motion. With the N2 gas at higher temperatures in the given problem, the molecules were energized to interact more strongly with their surroundings. This results in not just faster motion but more frequent collisions with the container's walls. That's why gas temperature is such a vital concept—it connects directly to a gas's energy dynamics.