Answer:
[tex]\frac{dT}{dt}=3.78^{\circ}K/min[/tex]
Step-by-step explanation:
We have to calculate the time derivative of T=PV/nR with P and V variable and n and R constants. This is:
[tex]\frac{dT}{dt} =\frac{d\frac{PV}{nR}}{dt}=\frac{1}{nR}\frac{d(PV)}{dt}[/tex]
What we have to do is the derivative of a product:
[tex]\frac{d(PV)}{dt}=P\frac{dV}{dt}+V\frac{dP}{dt}[/tex]
Substituting, we have:
[tex]\frac{dT}{dt} =\frac{P\frac{dV}{dt}+V\frac{dP}{dt}}{nR}[/tex]
where all these values are given since the time derivatives of P and V are their variation rate, using minutes.
We then substitute everything, noticing that already everything is in the same system of units so they cancel out:
[tex]\frac{dT}{dt}=\frac{P\frac{dV}{dt}+V\frac{dP}{dt}}{nR}=\frac{(8atm)(0.16L/min)+(13L)(0.14atm/min)}{(10mol)(0.0821Latm/mol^{\circ}K)}[/tex]
And then just calculate:
[tex]\frac{dT}{dt}=3.78^{\circ}K/min[/tex]
