Example 9.7: Measuring Temperature with a Volume Change
Temperature is sometimes measured with a gas thermometer by observing the change in the volume of the gas as the temperature changes at constant pressure. The hydrogen in a particular hydrogen gas thermometer has a volume of
150.0 cm^3 when immersed in a mixture of ice and water (
0.00 °C). When immersed in boiling liquid ammonia, the volume of the hydrogen, at the same pressure, is
131.7 cm^3. Find the temperature of boiling ammonia on the kelvin and Celsius scales.
Solution
\(V_{\mathrm{1}}\) \(= 150.0\ \mathrm{cm}^{3}\)
\(T_{\mathrm{1}}\) \(= 0.00\ \mathrm{°aC}\)
\(V_{\mathrm{2}}\) \(= 131.7\ \mathrm{cm}^{3}\)
\(T_{\mathrm{2}}\) = ?
A volume change caused by a temperature change at constant pressure means we should use Charles’s law. Taking
\(\ce{V1}\) and
\(T_{\mathrm{1}}\) as the initial values,
\(T_{\mathrm{2}}\) as the temperature at which the volume is unknown and
\(\ce{V2}\) as the unknown volume we have:
\(\dfrac{V_{\mathrm{1}}}{T_{\mathrm{1}}} = \dfrac{V_{\mathrm{2}}}{T_{\mathrm{2}}}\)
\(V_{\mathrm{1}} \cdot T_{\mathrm{2}} = V_{\mathrm{2}} \cdot T_{\mathrm{1}}\)
which means that
\(T_{\mathrm{2}}\) \(= T_{\mathrm{1}} \cdot \dfrac{V_{\mathrm{2}}}{V_{\mathrm{1}}}\)
\(\ \ \ =0.00\ \mathrm{°aC} \cdot \dfrac{131.7\ \mathrm{cm}^{3}}{150.0\ \mathrm{cm}^{3}}\)
\(\ \ \ =273.15\ \mathrm{K} \cdot 0.87800\)
\(\ \ \ =239.8\ \mathrm{K}\)
\(T_{\mathrm{2}}\) \(= 239.8\ \mathrm{K}\)
\(\ \ \ =-33.3\ \mathrm{°aC}\)