The temperature of 7.10 mol of an ideal monatomic gas is raised 15.0 K in an adiabatic process. What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change ÄEint in internal energy of the gas, and (d) the change ÄK in the average kinetic energy per atom?
Start with part c). In case of an ideal gas, the internal energy only depends on temperature. In case of a monoatomic gas we have:
E = 3/2 N k T
here k is Boltzmann's constant and N is given as
N = 7.1 mol = 4.276*10^24
So, we have:
E = 88.55 J/K T
If the temperature is raised by 15 , then the increase in E is thus 1328 J.
The change in internal eenrgy is also equal to the heat abosorbed minus the work done by the gas. An adiabatic change means, by definition, that the absorbed heat is zero. So, the work done by the gas is minus 1328 J.
To solve this problem, we need a few equations and concepts from thermodynamics:
1. The work done by a gas in an adiabatic process is given by the equation:
W = -(γ / (γ - 1)) * (P_final * V_final - P_initial * V_initial)
Here, γ is the adiabatic index or heat capacity ratio, which is 5/3 for a monatomic gas.
2. The energy transferred as heat (Q) in an adiabatic process is zero. This is because adiabatic processes occur without the transfer of heat.
3. The change in internal energy of a gas is given by:
ΔE_int = ΔQ - ΔW
As mentioned earlier, ΔQ is zero for an adiabatic process.
4. The average kinetic energy per atom is proportional to the temperature of the gas:
ΔK = (3/2) * R * ΔT
Here, R is the ideal gas constant and ΔT is the change in temperature.
Now, let's calculate the individual values:
(a) The work done by the gas (W):
Here, we need to know the initial and final volumes of the gas, denoted as V_initial and V_final, and the initial and final pressures, denoted as P_initial and P_final.
Without this information, it is not possible to calculate the work done by the gas.
(b) The energy transferred as heat (Q):
As mentioned earlier, in an adiabatic process, Q is zero. Therefore, Q = 0.
(c) The change in internal energy (ΔE_int):
Since Q = 0 in an adiabatic process, ΔE_int = -W. We will need the value of W calculated in part (a) to determine this.
(d) The change in average kinetic energy per atom (ΔK):
ΔK = (3/2) * R * ΔT
Here, R is the ideal gas constant (8.314 J/(mol*K)) and ΔT is the change in temperature (15.0 K).
Substituting the values, we can determine ΔK.
Therefore, we need additional information, such as the initial and final volumes or pressures, in order to calculate the work done by the gas and the change in internal energy. The energy transferred as heat is zero in an adiabatic process. However, we can calculate the change in average kinetic energy per atom using the equation mentioned above.