A sample of gas at P = 1123 Pa, V = 1.41 L, and T = 291 K is confined in a cylinder
a) Find the new pressure if the volume is reduced to half of the original volume at the same temperature.
b) If the temperature is raised to 391 K in the process of part (a), what is the new pressure?
c) If the gas is then heated to 582 K from the initial value and the pressure of the gas becomes 3369 Pa, what is the new volume?
Use P*V = n R T , the perfect gas law to solve these problems.
a) At constant T, note that P is inversely proportional to V
b) Use the gas law again, with the higher P = P2 from part a, and T = 391 K.
P2 = (V1/V2)*(T2/T1)
c) V3 = V2*(P2/P3)*(T3/T2)
the kinetic-molecular theory suggest that
To solve these problems, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure of the gas
V = volume of the gas
n = number of moles of the gas
R = gas constant (which is approximately 8.314 J/(mol*K))
T = temperature of the gas in Kelvin
a) To find the new pressure if the volume is reduced to half of the original volume at the same temperature, we can use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional at constant temperature.
Let's assume the initial pressure is P1, and the initial volume is V1. The final pressure (P2) after the volume becomes half (V2) would be:
P1 * V1 = P2 * V2
Substituting the given values:
P1 = 1123 Pa
V1 = 1.41 L
V2 = V1/2 = 1.41 L / 2 = 0.705 L
To find P2, rearrange the equation:
P2 = P1 * V1 / V2
P2 = 1123 Pa * 1.41 L / 0.705 L
P2 ≈ 2255.56 Pa
Therefore, the new pressure is approximately 2255.56 Pa.
b) If the temperature is raised to 391 K in the process of part (a), we can now use Charles's Law, which states that the volume and temperature of a gas are directly proportional at constant pressure.
Let's assume the initial temperature is T1, and the final temperature is T2. The new pressure (P3) after the temperature becomes T3 would be:
P2 * V2 / T2 = P3 * V2 / T3
Substituting the known values from previous calculations:
P2 = 2255.56 Pa
T2 = 291 K
T3 = 391 K
To find P3, rearrange the equation:
P3 = (P2 * T3) / T2
P3 = (2255.56 Pa * 391 K) / 291 K
P3 ≈ 3026.22 Pa
Therefore, the new pressure is approximately 3026.22 Pa.
c) If the gas is then heated to 582 K from the initial value and the pressure of the gas becomes 3369 Pa, we can now solve for the new volume (V3) using the ideal gas law equation.
PV = nRT
Let's assume the initial pressure is P1, the initial temperature is T1, and the final temperature is T3. The final volume (V3) would be:
P1 * V1 / T1 = P3 * V3 / T3
Substituting the known values from previous calculations:
P1 = 1123 Pa
V1 = 1.41 L
T1 = 291 K
P3 = 3369 Pa
T3 = 582 K
To find V3, rearrange the equation:
V3 = (P1 * V1 * T3) / (P3 * T1)
V3 = (1123 Pa * 1.41 L * 582 K) / (3369 Pa * 291 K)
V3 ≈ 1.98 L
Therefore, the new volume is approximately 1.98 L.