# Application of Derivatives

#### racuna

##### New member
The gas law for an ideal gas a absolute temperature T (in Kelvins), pressure P (in atmospheres), and volume V (in liters) is PV=nRT, where n is the number of moles of the gas and R=0.0821 is the gas constant. Suppose that, at a certain instant, P=7 atm and is increasing at a rate of 0.15 atm/min. and V=10L and is decreasing at a rate of 0.15L/min. Find the rate of change of T with respect to time at that instant if n=10 moles.

So, since the equation is PV=nRT do I solve for V and differentiate implicitly? Then what happens to all those numbers? I know I have to plug some in, but I don't know which ones and where because of the derivative.

Help? #### Unco

##### Senior Member
You have a function of two variables. Are you familiar with chain rule for a function of two variables?

$$\displaystyle \frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt}$$

Crikey I hope you have good eye sight to see that!

or
$$\displaystyle dx/dt = f_x (dx/dt) + f_y (dy/dt)$$

I'm pretty sure this is necessary :?.

#### wjm11

##### Senior Member
The gas law for an ideal gas a absolute temperature T (in Kelvins), pressure P (in atmospheres), and volume V (in liters) is PV=nRT, where n is the number of moles of the gas and R=0.0821 is the gas constant. Suppose that, at a certain instant, P=7 atm and is increasing at a rate of 0.15 atm/min. and V=10L and is decreasing at a rate of 0.15L/min. Find the rate of change of T with respect to time at that instant if n=10 moles.

So, since the equation is PV=nRT do I solve for V and differentiate implicitly? Then what happens to all those numbers? I know I have to plug some in, but I don't know which ones and where because of the derivative.
Racuna,

Rearrange the eqn to solve for T:

T = PV/nR

Recognize that this is a Related Rates problem, and rate means “something changing with respect to time.” Therefore, you want to differentiate the eqn implicitly with respect to time:

dT/dt = (1/nR)(P*dV/dt + V*dP/dt)

Now you simply plug in the appropriate values.

#### Unco

##### Senior Member
Oh good, that is the chain rule for a function of two variables .