Question

For each of the following pairs of gas properties, describe the relationship between the properties, describe a simple system that could be used to demonstrate the relationship, and explain the reason for the relationship:

(a) volume and pressure when number of gas particles and temperature are constant,

(b) pressure and temperature when volume and the number of gas particles are constant,

(c) volume and temperature when pressure and the number of gas particles are constant,

(d) the number of gas particles and pressure when volume and temperature are constant, and (e) the number of gas particles and volume when pressure and temperature are constant.

Answer 1

(a)

Consider a cylinder with a movable piston fitted at the top containing a certain number of gas particle at a constant temperature.

Now, if we pull the piston upward, the volume inside the cylinder will increase. Since the temperature and number of particles is constant, the particles will redistribute themselves over the increased volume. It will result in less number of particles hitting a unit area of the wall of the cylinder. Hence, the pressure, which is basically force per unit area on the wall will decrease.

Similarly, if we move the piston downward, the volume occupied by the gas particles will decrease, hence, there will be more number of particles hitting a unit area on the wall of the cylinder. Hence, the force per unit area on the wall applied by the gas particles is more. Hence, pressure has increased.

Hence, when volume increases, pressure decreases and vice versa. Hence, pressure and volume are inversely related at a constant temperature and number of particles.

Hence, the relationship between Pressure, P and Volume, V can be written as

The constant of proportionality will depend on the temperature T and number of particles, n.

(b)

Consider a closed vessel with a fixed number of gas particles inside it.

Now, lets heat the vessel from outside, which will increase its temperature. When the temperature increases, the gas particles will gain more kinetic energy and hence will hit the boundary walls more frequently. Hence, the force on the walls per area per area per time will be more. Hence, pressure will increase.

Similarly, if an ice bath is placed in contact with the vessel, the temperature of the gas inside the vessel will decrease. Hence, the gas particles will lose kinetic energy and hit the walls of the vessel less frequently. Hence, the pressure will decrease.

Hence, pressure increases with increased temperature.

Hence, we can write

Hence, ratio of P and T is constant at constant volume and number of particles.

(c)

Consider a cylinder with a movable piston fitted at the with constant pressure in the inside and a constant number of gas particles.

Now, if we increase the temperature of the gas particles inside the cylinder by heating, the kinetic energy of the gas molecules will increase. Now, since pressure is constant, the volume of the cylinder needs to increase to accommodate the extra force exerted by the molecules hitting the walls. Hence, the volume will increase by moving the piston upwards.

Similarly, if the cylinder is cooled down by decreasing the temperature, the kinetic energy of the gas particles will decrease. But to maintain the same pressure, the volume of the cylinder also needs to decrease. Hence, the piston will move downward and the volume will decrease.

Hence, when temperature increases, volume increases.

Hence, we can write

(d)

Consider a flask at constant volume and temperature.

We know that pressure inside the flasks comes from the the rate at which force is being applied on the wall of the flasks by the gas particles inside the flasks. Note that volume and temperature are constant. Hence, less the number of particles inside the flask, less is the pressure.

Hence, we can write

Hence, pressure is directly proportional to the number of particles at a constant temperature and volume.

(e)

Consider a cylinder fitted with a movable piston at constant pressure and temperature.

Since pressure and temperature are constant, increasing the number of gas particles will result in increased pressure, hence the volume must increase to maintain a constant pressure.

Hence, Volume and number of particles are directly proportional to each other at constant temperature and pressure.

Hence, we can write

Hence, the ratio of V and n is a constant at constant temperature and pressure.

From the above relationships, we see that both Volume and pressure are directly proportional to number of particles and
temperature.