Question

A college student is working on her physics homework in her dorm room. Her room contains a total of 7.0 ×1026 gas molecules. As she works, her body is converting chemical energy into thermal energy at a rate of 110 Watt .

PART A If her dorm room were an isolated system (dorm rooms can certainly feel like that) and if all of this thermal energy were transferred to the air in the room, by how much would the temperature increase in 10 min ? Use the ideal gas mode ANSWER IN Celsius please

Answer 1

Internal energy change of an ideal gas:
deltaU = n*cv*deltaT

First law of thermodynamics:
Q = deltaU + W

The only source of Q is her human body. W is 0 because the room is constant volume.

Thus:
Q = n*cv*deltaT

What is the molar isochoric specific heat of an ideal gas?
cv = R/(k - 1)

What is k? The adiabatic index. For air, oxygen and nitrogen, k=1.4. (I hope she is immersed in air, and not suffocated in pure Argon or pure CO2)

Thus"
Q = n*R*deltaT/(k - 1)

Solve for deltaT:
deltaT = Q*(k - 1)/(n*R)

Number of moles in terms of number of molecules:
n = N/N_A

Total heat in terms of heat flow and time:
Q = Q_dot*t

Thus:
deltaT = Q_dot*(k - 1)*N_A*t/(N*R)

Data:
Q_dot:=110 Watts; t:=600 sec; k:=1.4; N:=7.0e26 molecules; R:=8.314 J/mol-K; N_A:=6.022e23 items/mole;

Result:
deltaT = 2.732 Kelvin

Ok, I can answer in Celsius with no effort. Celsius and Kelvin are by definition the same "size of the degrees", and thus temperature changes in C and K are identical.

deltaT = 2.732 Celsius