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?
To solve this problem, we need to use some concepts from thermodynamics, specifically the equations for work, heat, internal energy, and average kinetic energy.
(a) The work done by the gas in an adiabatic process can be determined using the following equation:
W = -nCvΔT
where W represents work, n represents the number of moles of gas, Cv represents the molar specific heat capacity at constant volume, and ΔT represents the change in temperature. In this case, n = 7.10 mol and ΔT = 15.0 K.
To find the molar specific heat capacity at constant volume, we can use the ideal gas equation:
PV = nRT
Since the process is adiabatic, there is no heat transfer and therefore no change in pressure or volume. This means that the molar specific heat capacity at constant volume, Cv, is given by:
Cv = (3/2)R
where R is the gas constant.
Now we can substitute the values into the equation for work:
W = -nCvΔT
W = -(7.10 mol)(3/2)R(15.0 K)
(b) Since the process is adiabatic, no heat is transferred into or out of the system. Therefore, Q = 0.
(c) The change in internal energy (ΔEint) of the gas can be determined using the first law of thermodynamics:
ΔEint = Q - W
Since Q = 0 (as mentioned in part (b)), the equation becomes:
ΔEint = -W
So ΔEint is equal to the negative value of the work calculated in part (a).
(d) The change in average kinetic energy per atom (ΔK) can be determined using the following equation:
ΔK = (3/2)nRΔT
Substituting the given values:
ΔK = (3/2)(7.10 mol)R(15.0 K)
Now you can calculate the values for (a), (c), and (d) using the equations and given values. Remember to use appropriate unit conversions if necessary.