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Chapter 10: Problem 79

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

Expert verified
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

01

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.
02

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): \(250 + 273.15 = 523.15\,\text{K}\)
03

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. $$\frac{K.E._{final}}{K.E._{initial}} = \frac{T2}{T1}$$
04

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.
05

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.
06

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.

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Most popular questions from this chapter

A 6.0-L tank is filled with helium gas at a pressure of 2 MPa. How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of $101.3 \mathrm{kPa}$, assuming that the temperature remains constant and that the tank cannot be emptied below \(101.3 \mathrm{kPa}\) ?

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