#### To determine

**To find:** The rate of change of *T* with respect to time if *n* = 10 mol.

#### Answer

The rate of change of *T* with respect to time if *n* = 10 mol is approximately −0.2436 K/min.

#### Explanation

**Given:**

The gas law for an ideal gas is given by *PV* = *nRT*, where *T* is absolute temperature in Kelvins, *P* is pressure in atmosphere, *V* is volume in liters, *n* is the number of moles of the gas and the value of gas constant *R* = 0.0821.

Also given that *P* = 8 atm and dPdt=0.10 atm/min and *V* = 10 L, dVdt=−0.15 atm/min

**Calculation:**

An ideal gas function is given by *PV* = *nRT*.

Express the function in terms of T as, T=PVnR.

Substitute, *R* = 0.0821 and *n* = 10,

T=PV(10)(0.0821)=PV0.821

Obtain the derivative of *T* with respect to *t* as follows,

dTdt=10.821[P(t)⋅V'(t)+V(t)⋅P'(t)] (1)

Substitute the respective values in the equation (1),

dTdt=10.821[8(−0.15)+10(0.1)]=10.821[−1.2+1]=10.821[−0.2]≈−0.2436

Thus, the derivative, dTdt=−0.2436 K/min.