A piston contains 1m^3 of helium at a temperature of 250 K and an absolute pressure of 1.0kPa. The following changes are made in sequence:
(a) First the volume of the piston is decreased to 0.2 m^3 while the temperature is kept constant, what is the pressure of the gas?
(b) Taking the gas after process (a) has been carried out. Now the volume of the gas is kept fixed at 0.2 m^3 but the temperature of the gas is raised to 500K. What does the pressure of the gas become?
(a) At constant T, P*V is constant.
(b) At constant V, P is proportional to (absolute) T. T goes from 250K to 500 K in this step.
To solve this problem, we can use the ideal gas law equation, which states:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
(a) In this scenario, the volume decreases while the temperature is kept constant. According to the ideal gas law, if the volume decreases while the temperature is constant, the pressure will increase. We can calculate the pressure of the gas using the following steps:
1. Given:
Volume (V1) = 1 m^3
Temperature (T1) = 250 K
Pressure (P1) = 1.0 kPa
2. We need to find the pressure after the volume is decreased to 0.2 m^3 while the temperature is kept constant.
Volume (V2) = 0.2 m^3
3. Since temperature remains constant, we can use the formula PV = nRT to find the pressure.
P1V1 = P2V2
P2 = (P1V1) / V2
= (1.0 kPa * 1 m^3) / 0.2 m^3
= 5.0 kPa
Therefore, the pressure of the gas after the volume is decreased to 0.2 m^3 while the temperature is kept constant is 5.0 kPa.
(b) In this scenario, the volume is kept fixed at 0.2 m^3, but the temperature of the gas is raised to 500 K. To find the new pressure, we need to use the ideal gas law equation again:
1. Given:
Volume (V2) = 0.2 m^3
Temperature (T1) = 250 K
2. We need to find the pressure of the gas after the temperature is raised to 500 K.
Temperature (T2) = 500 K
3. Since the volume remains constant, we can again use the formula PV = nRT to find the pressure.
P1V1 / T1 = P2V2 / T2
P2 = (P1V1 * T2) / (V2 * T1)
= (1.0 kPa * 1 m^3 * 500 K) / (0.2 m^3 * 250 K)
= 10 kPa
Therefore, the pressure of the gas after the volume is kept fixed at 0.2 m^3 and the temperature is raised to 500 K is 10 kPa.