# The Question

*The pressure, volume, and temperature of an ideal gas are related by the equation (*when pressure is measured in kilopascals*). Find the rate at which the pressure is changing when the temperature is 300 K and increasing at a rate of 0.1 K s , and the volume is 100 L and increasing at a rate of 0.2 L s.*

# Solution

First of all, solving for *:*

*.*

Now we can go further and say that both and are functions of time, . So we have:

.

Now changes to pressure, are going to result from independent changes to temperature, , and pressure .

Now changes to temperature and volume are due to changes in time, . It can be shown that this change manifests itself as (with the obvious generalisation to derivatives of functions of the form ):

.

(See: http://library.ucc.ie/search~S0?/Xcalculus%20by%20stewart&SORT=D/Xcalculus%20by%20stewart&SORT=D&SUBKEY=calculus%20by%20stewart/1,3576,3576,B/frameset&FF=Xcalculus%20by%20stewart&SORT=D&3,3, for details of this.)

Now to calculate this rate of change at we must put these values into , , so:

.

Since we know each of the constituents of this equation (, ) all that is left is to do the calculations.

at .

.

Hence:

kPa.

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