First-order thermal isolation ($C\dot{\delta T_s}+G\delta T_s=G\delta T_b+\delta P$)

A lumped sample of heat capacity $C$ coupled by thermal conductance $G$ to a bath has the displayed linearized energy balance. For constant internal power, the bath-to-sample temperature response is $1/(1+i\omega C/G)$. Insulation and heat capacity suppress rapid bath fluctuations but have unity static bath-to-sample gain. A steady power change shifts the sample by $\delta P/G$, so improved isolation increases self-heating sensitivity and equilibration time. The lumped approximation requires internal equilibration faster than the variations being modeled.

A thermal time constant characterizes exponential relaxation in a first-order lumped thermal model. For heat capacity $C$ and thermal conductance $G$ to a fixed bath, a small temperature deviation obeys $C\dot{\delta T}=-G\delta T$, givingAfter one $\tau$, the initial deviation has fallen to $1/e$ of its original value. The ratio has units of time because $C$ is measured in $\mathrm{J}{\mathrm{K}}^{-1}$ and $G$ in $\mathrm{W}{\mathrm{K}}^{-1}$.

A heating step provides an experimental check: the body approaches its final temperature rise with the same $\tau$ in this linear model. Greater heat capacity slows changes, while greater thermal conductance speeds exchange. Several coupled regions generally produce several relaxation modes, so fitting one exponential over a short interval does not establish a unique time constant for the entire apparatus.