A college student is working on her physics homework in her dorm room. Her room

Question

A college student is working on her physics homework in her dorm room. Her room contains a total of 6.0×1026 gas molecules. As she works, her body is converting chemical energy into thermal energy at a rate of 125 W.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 15 min ? Use the ideal gas model. Express your answer using two significant figures.

Explanation / Answer

here,

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

specific heat of an ideal gas

cv = R/(k - 1)

k is adiabatic index.

For air, oxygen and nitrogen, k = 1.4

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)

Q_dot:= 125 Watts

t = 900 sec;

k = 1.4

N = 6.0 * 10^26 molecules;

R:=8.314 J/mol-K;

N_A: = 6.022 * 10^23 items/mole

then,

deltaT = 125 * (1.4 - 1)* 6.022 * 10^23 * 900 / ( 6 * 10^26 * 8.314)

delta T = 5.43 K

the increase in temprature is 5.43 K

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