# The rate of change of T with respect to time if n = 10 mol.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.8, Problem 35E
To determine

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

Expert Solution

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

### Explanation of Solution

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=0.10atm/min and V = 10 L, dVdt=0.15atm/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=0.2436K/min.

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