Suppose that a closed system with constant number of moles of a gas and a constant volume has pressure 205 kN/m^2 when the temperature is 13 Celsius. What Celsius temperature will result in a pressure of 239.8 kN/m^2?
The absolute temperature T is (273+ 13 ) K = 286 K. Note that it is always essential to use absolute temperatures with the gas laws.
Since n and V are constant, we algebraically rearrange PV=nRT to the form P/T = nR/V. Since the right-hand side is now constant (since each factor on this side is constant), we see that P/T is constant.
Thus, when T increases, P must increase in the same proportion (otherwise P/T would change).
If, as in the present example, P changes from 205 kN/m^2 to 239.8 kN/m^2; the proportional change is
The new temperature will therefore be
If PV = n R T, then if n and V are constant so is the ratio P / T. Since then P1 / T1 = P2 / T2, P2 = P1 * (V2 / V1).
A more intuitive way of looking at this is to realize that whenever P / T is constant, it follows that if the temperature changes from T1 to T2 while pressure changes from P1 to P2, we must have (P2 / P1 ) = (T2 / T1). That is, the pressure and volume ratios are equal.
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