There is hydrogen with a mass of m=40 g, which was at a temperature of T=300 K. The gas expanded adiabatically, increasing the volume by n1=3 times. Then the gas was isothermally compressed to volume, reducing it by n2=2 times. It is necessary to determine the total work A performed by the gas and its final temperature T.
Answer:
First, let's find the initial gas pressure. To do this, we use the equation of state of an ideal gas:
pV = nRT,
where p is the gas pressure, V is its volume, n is the amount of gas substance, R is the universal gas constant, T is the gas temperature.
The amount of substance in a gas can be found by dividing the mass by the molar mass:
n = m/M,
where M is the molar mass of the gas. For hydrogen M = 2 g/mol.
Then the initial gas pressure is:
p1 = (m/M)RT/V = (40 g)/(2 g/mol) * 8.31 J/(mol*K) * 300 K / (1 l) = 4.99 * 10^5 Pa .
Next, we will find the work done by the gas during adiabatic expansion. Since the process is adiabatic, then Q = 0, and the first law of thermodynamics takes the form:
dU = -pdV,
where dU is the change in the internal energy of the gas, p and V are the pressure and volume of the gas, respectively.
Since the process is adiabatic, dU = Cv*dT, where Cv is the heat capacity of the gas at constant volume.
Then:
Cv*dT = -pdV,
CvdT/T = -pdV/(TV),
integrating this expression from the initial temperature and volume to the final values, we obtain:
ln(T2/T1) = -ln(V2/V1) * (Cv/R),
where T2 is the final temperature of the gas, V2 is its volume after adiabatic expansion.
The heat capacity of a gas at constant volume can be found from the relation:
Cp - Cv = R,
where Cp is the heat capacity of gas at constant pressure. For an ideal gas Cp = Cv + R.
Then:
Cv = Cp - R = 7/2 R.
After adiabatic expansion, the gas volume became n1 = 3 times greater than the initial one, then the final volume:
V2 = n1 * V1 = 3 * V1.
Then, substituting all known values into the formula for ln(T2/T1), we find the final gas temperature:
T2 = T1 * (V1/V2)^((7/2)R) = 300 К * (1/3)^((7/2)*8,31/1000) = 219,6 К.
Next, we will find the work done by a gas under isothermal compression. Since the process is isothermal, then T = const, and the first law of thermodynamics takes the form:
dU = -pdV + Q = -pdV,
where Q is the heat received or given off by the gas.
Integrating this expression from the final volume to the initial volume, we obtain:
W = -∫p2^1 V dV,
where p2 is the final gas pressure after compression.
Using the equation of state of an ideal gas and the condition of isothermal process, we find the final gas pressure:
p2 = p1 * (V1/V2) = p1 * (n1/n2),
where n2 is the final amount of gas substance after compression.
Then the work of gas under isothermal compression:
W = -∫p2^1 V dV = -∫(p1 * (n1/n2))^p1 (n2/n1 * V1)^2/3 d((n2/n1 * V1)^2/3) = -p1 * V1 * (n1/n2) * [(n2/n1)^2/3 - 1],
where we used the relationship between V and n for an ideal gas in an isothermal process: nV = const.
Then the total work done by the gas is:
A = W1 + W2 = -p1 * V1 * (n1/n2) * [(n2/n1)^2/3 - 1],
where W1 is the work done by the gas during adiabatic expansion, W2 is the work done by the gas during isothermal compression.
Substituting the known values, we get:
A = -4.99 * 10^5 Pa * 1 l * (3/2) * [(2/3)^2/3 - 1] = 5.02 * 10^4 J.
And of course, the final gas temperature after going through both processes is T2 = 219.6 K.
Thus, we have found the total work done by the gas and its final temperature after adiabatic expansion and isothermal compression.
The digital goods store presents a digital product - calculation material for a problem on the topic of thermodynamics.
This material examines the process of adiabatic expansion and isothermal compression of hydrogen with a mass of m = 40 g, which had an initial temperature of T = 300 K.
The calculation material contains a detailed description of the conditions of the problem, the formulas and laws used, the derivation of the calculation formula and answers to the questions posed in the problem.
Product Description: The digital goods store provides calculation material for a problem on the topic of thermodynamics. This material examines the process of adiabatic expansion and isothermal compression of hydrogen with a mass of m = 40 g, which had an initial temperature of T = 300 K. The calculation material contains a detailed description of the conditions of the problem, the formulas and laws used, the derivation of the calculation formula and answers to the questions posed in the problem.
Task: Hydrogen with a mass of m = 40 g, which had a temperature of T = 300 K, expanded adiabatically, increasing its volume by n1 = 3 times. Then, during isothermal compression, the gas volume decreased by n2=2 times. Determine the total work A performed by the gas and the final temperature T of the gas. Problem 20046.
Solution: First, let's find the initial gas pressure. To do this, we use the equation of state of an ideal gas:
pV = nRT,
where p is the gas pressure, V is its volume, n is the amount of gas substance, R is the universal gas constant, T is the gas temperature.
The amount of substance in a gas can be found by dividing the mass by the molar mass:
n = m/M,
where M is the molar mass of the gas. For hydrogen M = 2 g/mol.
Then the initial gas pressure is:
p1 = (m/M)RT/V = (40 g)/(2 g/mol) * 8.31 J/(mol*K) * 300 K / (1 l) = 4.99 * 10^5 Pa .
Next, we will find the work done by the gas during adiabatic expansion. Since the process is adiabatic, then Q = 0, and the first law of thermodynamics takes the form:
dU = -pdV,
where dU is the change in the internal energy of the gas, p and V are the pressure and volume of the gas, respectively.
Since the process is adiabatic, dU = Cv*dT, where Cv is the heat capacity of the gas at constant volume.
Then:
Cv*dT = -pdV,
CvdT/T = -pdV/(TV),
integrating this expression from the initial temperature and volume to the final values, we obtain:
ln(T2/T1) = -ln(V2/V1) * (Cv/R),
where T2 is the final temperature of the gas, V2 is its volume after adiabatic expansion.
The heat capacity of a gas at constant volume can be found from the relation:
Cp - Cv = R,
where Cp is the heat capacity of gas at constant pressure. For an ideal gas Cp = Cv + R.
Then:
Cv = Cp - R = 7/2 R.
After adiabatic expansion, the gas volume became n1 = 3 times greater than the initial one, then the final volume:
V2 = n1 * V1 = 3 * V1.
Then, substituting all known values into the formula for ln(T2/T1), we find the final gas temperature:
ln(T2/T1) = -ln(3) * (7/2) = -2,303 * (7/2) = -8,058,
T2/T1 = e^(-8,058) = 0,000329,
T2 = T1 * 0.000329 = 300 K * 0.000329 = 0.0987 K.
Now let's find the work done by a gas under isothermal compression. Since the process is isothermal, then T = const, and the first law of thermodynamics takes the form:
dU = Q - pdV,
where Q is the heat transferred to the gas, dU is the change in the internal energy of the gas.
Since the process is isothermal, T = const, therefore, Q = W, that is, the work done by the gas is equal to the heat transferred to the gas.
Then:
W = Q = nRT * ln(V1/V2),
where V1 and V2 are the initial and final volumes of gas, respectively.
After adiabatic expansion, the gas volume became n1 = 3 times greater than the initial one, and then, during isothermal compression, the gas volume decreased by n2 = 2 times. Then the final volume of gas is:
V2 = V1 * (1/n2) = V1/2.
Then the gas work:
W = nRT * ln(V1/(V1/2)) = nRT * ln(2) = (40 g)/(2 g/mol) * 8.31 J/(mol*K) * 300 K * ln( 2) = -4986.54 J.
Answers to the questions posed in the problem:
The total work done by a gas during adiabatic expansion and isothermal compression is W = -4986.54 J.
The final gas temperature after adiabatic expansion and isothermal compression is T2 = 0.0987 K.
***
Product description:
This product is a sample of hydrogen weighing m=40 g, which had a temperature T=300 K. Next, the gas was adiabatically expanded, increasing the volume by n1=3 times. Then isothermal compression of the gas occurred, as a result of which the volume decreased by n2=2 times.
To determine the total work A performed by the gas and the final temperature T of the gas, you can use the Mayer equation:
A = C_v * (T_2 - T_1) + C_p * (T_2 - T_1)
where C_v and C_p are specific heat capacities at constant volume and constant pressure, respectively, T_1 and T_2 are the initial and final gas temperatures.
For hydrogen, specific heat capacities can be calculated using the formulas:
C_v = (3/2) * R C_p = (5/2) * R
where R is the universal gas constant.
Thus, the total work A will be equal to:
A = (3/2) * R * (T_2 - T_1) + (5/2) * R * (T_2 - T_1)
To determine the final gas temperature T, the following relationship can be used:
T_2 = T_1 * (n1/n2)^((C_p - C_v)/C_p)
where n1 and n2 are the coefficients of change in gas volume during adiabatic expansion and isothermal compression, respectively.
Substituting the data from the problem statement, we get:
T_2 = 300 * (3/2)^((5/2 - 3/2)/(5/2)) * (1/2) = 150 K
A = (3/2) * R * (150 - 300) + (5/2) * R * (150 - 300) = -600 R Дж
Thus, the total work done by the gas is -600 R J, and the final temperature of the gas is 150 K.
***
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