Question:
Air is pumped into an automobile tube upto a pressure of 200 kPa in the morning when the air temperature is 22$^{\circ}$C. During the day, temperature rises to 42$^{\circ}$C and the tube expands by 2 %. The pressure of the air in the tube at this temperature, will be approximately
Air is pumped into an automobile tube upto a pressure of 200 kPa in the morning when the air temperature is 22$^{\circ}$C. During the day, temperature rises to 42$^{\circ}$C and the tube expands by 2 %. The pressure of the air in the tube at this temperature, will be approximately
Updated On: Jun 20, 2022
- 212 kPa
- 209 kPa
- 206 kPa
- 200 kPa
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The Correct Option is B
Solution and Explanation
The ideal gas law is the equation of state of an ideal gas. The state of an amount of gas is determined by its pressure, volume and temperature. The equation has the form pV = nRT
where, p is pressure, V the volume, n the number of moles, R the gas constant and T the temperature
$\therefore \, \, \, \, \, \, \frac{p_1 V_1}{T_1} = \frac{p_2V_2}{T_2}$
Given $ \, P_1 = 200 kpa, V_1,= T_1 = 273 + 22 = 295 K,$
$V_2 = V + 0.02V , T_2 = 273 + 42 = 315 K$
$ \, \, \, \, \, \, \frac{200 \times V}{295} = \frac{p_2 \times 1.02V}{315}$
$\Rightarrow \, \, \, \, \, \, \, \, \, p_2 = \frac{200 \times 315}{295 \times 1.02}$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, p_2 \approx 209 kPa$
where, p is pressure, V the volume, n the number of moles, R the gas constant and T the temperature
$\therefore \, \, \, \, \, \, \frac{p_1 V_1}{T_1} = \frac{p_2V_2}{T_2}$
Given $ \, P_1 = 200 kpa, V_1,= T_1 = 273 + 22 = 295 K,$
$V_2 = V + 0.02V , T_2 = 273 + 42 = 315 K$
$ \, \, \, \, \, \, \frac{200 \times V}{295} = \frac{p_2 \times 1.02V}{315}$
$\Rightarrow \, \, \, \, \, \, \, \, \, p_2 = \frac{200 \times 315}{295 \times 1.02}$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, p_2 \approx 209 kPa$
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Concepts Used:
Kinetic Molecular Theory of Gases
Postulates of Kinetic Theory of Gases:
- Gases consist of particles in constant, random motion. They continue in a straight line until they collide with each other or the walls of their container.
- Particles are point masses with no volume. The particles are so small compared to the space between them, that we do not consider their size in ideal gases.
- Gas pressure is due to the molecules colliding with the walls of the container. All of these collisions are perfectly elastic, meaning that there is no change in energy of either the particles or the wall upon collision. No energy is lost or gained from collisions. The time it takes to collide is negligible compared with the time between collisions.
- The kinetic energy of a gas is a measure of its Kelvin temperature. Individual gas molecules have different speeds, but the temperature and
kinetic energy of the gas refer to the average of these speeds. - The average kinetic energy of a gas particle is directly proportional to the temperature. An increase in temperature increases the speed in which the gas molecules move.
- All gases at a given temperature have the same average kinetic energy.
- Lighter gas molecules move faster than heavier molecules.
