Starting with volume and temperature of 9.2 x 10 m³ and 290 K, respectively, 0.183 [6] moles of an ideal gas contained in a cylinder with a moveable frictionless piston undergoes the following order of processes in a thermodynamic cycle: isothermal compression to a volume of 2.3 x 10 m³, isochoric heating to 420 K, isobaric expansion to 4.4 x 10 m³, and adiabatic cooling back to the original state.
a. Define each of the four processes that the gas goes through.
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b. Evaluate the missing state variables (P.V.T) at the end of each thermodynamic process.
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c. Neatly sketch a PV diagram for the cycle indicating actual values.
a. The four processes that the gas goes through in the thermodynamic cycle are:
1. Isothermal compression: In this process, the gas is compressed while maintaining a constant temperature. The gas volume decreases, but the pressure increases.
2. Isochoric heating: In this process, the gas is heated while the volume remains constant. The pressure and temperature of the gas increase.
3. Isobaric expansion: In this process, the gas expands while the pressure remains constant. The volume of the gas increases, and the temperature decreases.
4. Adiabatic cooling: In this process, the gas undergoes a decrease in temperature without any heat entering or leaving the system. The volume decreases, and the pressure decreases as well.
b. To evaluate the missing state variables at the end of each process, we use the ideal gas law equation:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
At the end of each process, we can calculate the missing state variables:
1. Isothermal compression:
Initial volume (V1) = 9.2 x 10 m³
Initial temperature (T1) = 290 K
Final volume (V2) = 2.3 x 10 m³
Using the ideal gas law equation:
P1V1/T1 = P2V2/T2
We can solve for P2:
P2 = (P1V1/T1) * (T2/V2)
P2 = (P1V1 * T2) / (T1 * V2)
2. Isochoric heating:
Pressure (P2) = Value obtained from the previous calculation
Initial temperature (T2) = Value obtained from the previous calculation
Final temperature (T3) = 420 K
Using the ideal gas law equation:
P2V2/T2 = P3V3/T3
We can solve for P3:
P3 = (P2V2/T2) * (T3/V3)
P3 = (P2V2 * T3) / (T2 * V3)
3. Isobaric expansion:
Pressure (P3) = Value obtained from the previous calculation
Final volume (V3) = 4.4 x 10 m³
Initial temperature (T3) = Value obtained from the previous calculation
Using the ideal gas law equation:
P3V3/T3 = P4V4/T4
We can solve for T4:
T4 = (P3V3/T3) * (T4/V4)
T4 = (P3V3 * T4) / (T3 * V4)
4. Adiabatic cooling:
Initial volume (V4) = 4.4 x 10 m³
Initial temperature (T4) = Value obtained from the previous calculation
Final temperature (T1) = 290 K
Using the ideal gas law equation:
P4V4/T4 = P1V1/T1
We can solve for P4:
P4 = (P1V1/T1) * (T4/V4)
P4 = (P1V1 * T4) / (T1 * V4)
c. To sketch the PV diagram for the cycle, we plot pressure (P) on the y-axis and volume (V) on the x-axis. The actual values obtained from the calculations in part b are then plotted on the diagram for each process.
a.
1. Isothermal compression: In this process, the temperature of the gas remains constant while the volume is compressed. The gas undergoes work as it is compressed, resulting in a decrease in volume.
2. Isochoric heating: Also known as isovolumetric or constant volume heating, this process occurs at constant volume. The gas is heated, resulting in an increase in temperature while the volume remains constant.
3. Isobaric expansion: In this process, the gas expands while maintaining a constant pressure. The volume of the gas increases while the temperature may change.
4. Adiabatic cooling: Adiabatic means no heat is exchanged with the surroundings. In this process, the gas is allowed to cool down without any heat transfer. The temperature of the gas decreases while the volume may change.
b.
To evaluate the missing state variables (P.V.T) at the end of each thermodynamic process, we can use the ideal gas law:
PV = nRT
where:
P = pressure
V = volume
n = moles of gas
R = ideal gas constant
T = temperature
1. Isothermal compression:
Given:
Initial volume (V1) = 9.2 × 10 m³
Initial temperature (T1) = 290 K
Final volume (V2) = 2.3 × 10 m³
Using the ideal gas law, we can calculate the final pressure (P2):
P1V1/T1 = P2V2/T2
2. Isochoric heating:
Given:
Initial volume (V2) = 2.3 × 10 m³
Initial temperature (T2) = T1 (final temperature from previous step)
Final temperature (T3) = 420 K
Since the volume remains constant, the pressure remains the same throughout this process.
3. Isobaric expansion:
Given:
Initial volume (V3) = V2 (final volume from previous step)
Initial temperature (T3) = 420 K
Final volume (V4) = 4.4 × 10 m³
Since the pressure remains constant, we can use the ideal gas law to calculate the final temperature (T4).
4. Adiabatic cooling:
Given:
Initial volume (V4) = 4.4 × 10 m³
Initial temperature (T4) = final temperature from previous step (T1)
Final volume (V1) = V1 (initial volume)
Using the ideal gas law, we can calculate the final pressure (P1).
c.
Unfortunately, as an AI text-based bot, I am unable to draw sketches or diagrams. However, I can describe the PV diagram for the cycle:
The PV diagram will have a series of steps:
1. From initial state point (V1, P1) to state point 2 (V2, P2) representing isothermal compression.
2. From state point 2 to state point 3 (V3, P2) representing isochoric heating.
3. From state point 3 to state point 4 (V4, P3) representing isobaric expansion.
4. From state point 4 back to the initial state point (V1, P4) representing adiabatic cooling.
Note that the actual values of volumes (V1, V2, V3, V4) and pressures (P1, P2, P3, P4) should be plotted on the PV diagram.