A closed system is sealed so that no gas can enter or leave the system. The system is held at constant temperature.
The system is observed to have a pressure of 87.7 kN/m^2 when its volume is 1.56 m^3. What is the pressure when the volume is increased to 3.12 m^3?.
We know that for an ideal gas PV = nRT.
If n and T are constant, then nRT must be constant and so must PV. If the volume increases by a certain factor, the pressure must therefore decrease by the same factor.
So when the volume increases from 1.56 to 3.12 m^3, a factor of
the pressure will decrease to
Alternatively we could say that since PV = constant, P1 V1 = P2 V2 so
Since PV = nRT for an ideal gas, V = nRT / P. If P changes from P1 to P2, V changes from V1 = n R T / P1 to V2 = n R T / P2; the volume ratio is
volume ratio = V2 / V1 = (n R T / P2) / (n R T / P1) = P1 / P2.
This proportionality shows that when other factors are constant, pressure and volume ratios are inverse to one another; as a result, e.g., a doubling of pressure results in a halving of volume, and vice versa.
To use this relationship we can invert the equation and solve P2 / P1 = V1 / V2 for P2 to obtain P2 = P1 * (V1 / V2)
Another way of seeing that pressure and volume are inversely proportional is to observe that when other factors are constant, n R T is constant (n and T being among the 'other factors', and R being always the same in every situation). It follows that PV must be constant, so that when either P or V increases the other must decrease by the same factor.
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