The two vessels are connected by a tube to a tap. One of them contains 1 kmol of hydrogen under a pressure of 4 MPa

| February 15, 2021 | education

The two vessels are connected by a tube to a tap. One of them contains 1 kmol of hydrogen under a pressure of 4 MPa at a temperature of 300 K, and the second is evacuated. After opening the valve, the pressure and temperature of hydrogen became equal to 2 MPa and, accordingly, 270 K. Determine: a) the mass of hydrogen and the volume of the first vessel b) the density of hydrogen before opening the valve c) the volume of the second vessel d) the ratio of the thermal velocities of hydrogen molecules before and after opening the tap

ν = 1 kmol = 1000 mol.
M (H2) = 2 * 10 ^ -3 kg / mol.
P1 = 4 MPa = 4 * 10 ^ 6 Pa.
T1 = 300 K.
R = 8.31 J / mol * K.
P2 = 2 MPa = 2 * 10 ^ 6 Pa.
T1 = 270 K.
m -?
V1 -?
V2 -?
ρ1 -?
(V1) -?
(V2) -?
The amount of substance ν is determined by the formula: ν = m / M, where m is the mass of the gas, M is the molar mass of the gas.
m = ν * M.
m = 1000 mol * 2 * 10 ^ -3 kg / mol = 2 kg.
Let’s write the Mendeleev-Cliperon equations for the first vessel: P1 * V1 = m * R * T1 / M.
V1 = m * R * T1 / M * P1.
V1 = 2 kg * 8.31 J / mol * K * 300 K / 2 * 10 ^ -3 kg / mol * 4 * 10 ^ 6 Pa = 0.623 m ^ 3.
Let us find the density of the gas in the first vessel ρ1 = m / V1.
ρ1 = 2 kg / 0.623 m ^ 3 = 3.2 kg / m ^ 3.
Let us write the Mendeleev-Cliperon equations for the vessels after their connection: P2 * V = m * R * T2 / M, where V is the volume of the two vessels together.
V = V1 + V2.
V2 = V – V1.
V = m * R * T2 / M * P2.
V = 2 kg * 8.31 J / mol * K * 270 K / 2 * 10 ^ -3 kg / mol * 2 * 10 ^ 6 Pa = 1.121 m ^ 3.
V2 = 1.121 m ^ 3 – 0.623 m ^ 3 = 0.498 m ^ 3.
The mean square velocity of the molecules is determined by the formula: (V) = √ (3 * R * T / M).
(V1) = √ (3 * R * T1 / M).
(V2) = √ (3 * R * T2 / M).
(V1) / (V2) = √ (3 * R * T1 / M) / √ (3 * R * T2 / M) = √T1 / √T2.
(V1) / (V2) = √300 K / √270 K = 1.05.
Answer: m = 2 kg, V1 = 0.623 m ^ 3, ρ1 = 3.2 kg / m ^ 3, V2 = 0.498 m ^ 3, (V1) / (V2) = 1.05.



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