Question 3.2: You have accidentally arrived at the end of the range of an ...
You have accidentally arrived at the end of the range of an ethanol-in-glass thermometer so that the entire volume of the glass capillary is filled. By how much will the pressure in the capillary increase if the temperature is increased by another 10.0°C? \beta_{glass} = 2.00× 10^{-5}\left( °C\right)^{-1}, \beta_{ethanol}= 11.2 × 10^{-4}\left( °C \right)^{-1}, and \kappa _{ethanol} =11.0 ×10^{-5}\left( bar \right)^{-1}. Will the thermometer survive your experiment?
Using Equation (3.11),
\Delta P=\int_{T_{i}}^{T_{f}}\frac{\beta}{\kappa }dT – \int_{V_{i}}^{V_{f}}\frac{1}{\kappa V}dV\approx \frac{\beta}{\kappa }\left( T_{f}-T_{i} \right)-\frac{1}{\kappa }\ln \frac{V_{f}}{V_{i}} (3.11)
\Delta P=\int_{}^{}\frac{\beta_{ethanol}}{\kappa }dT – \int_{}^{}\frac{1}{\kappa V}dV \approx \frac{\beta_{ethanol}}{\kappa } \Delta T -\frac{1}{\kappa }\ln \frac{V_{f}}{V_{i}}
=\frac{\beta_{ethanol}}{\kappa }\Delta T-\frac{1}{\kappa }\ln \frac{V_{i}\left( 1+\beta_{glass}\Delta T \right)}{V_{i}}\approx \frac{\beta_{ethanol}}{\kappa }\Delta T -\frac{1}{\kappa }\frac{V_{i}\beta_{glass}\Delta T}{V_{i}}
=\frac{\left( \beta_{ethanol}-\beta_{glass} \right)}{\kappa }\Delta T
=\frac{\left( 11.2-0.200 \right) × 10^{-4}\left( °C \right)^{-1}}{11.0 × 10^{-5}\left( bar \right)^{-1}} × 10.0 °C =100.bar
In this calculation, we have used the relations V\left( T_{2} \right)=V\left( T_{1} \right)\left( 1+\beta\left[ T_{2}-T_{1}\right] \right) and \ln\left( 1 +x \right)\approx x if x\ll 1. The glass is unlikely to withstand such a large increase in pressure.