JEE Main & Advanced Physics Transmission of Heat Cooling Curves

(1) Curve between \[log(\theta -{{\theta }_{0}})\] and time

As \[\frac{d\theta }{dt}\propto -(\theta -{{\theta }_{0}})\Rightarrow \,\frac{d\theta }{(\theta -{{\theta }_{0}})}=-Kdt\]

Integrating \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+C\] \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+{{\log }_{e}}A\]

This is a straight line with negative slope

(2) Curve between temperature of body and time

As  \[{{\log }_{e}}(\theta -{{\theta }_{0}})=-Kt+{{\log }_{e}}A\]\[\Rightarrow \]\[{{\log }_{e}}\frac{\theta -{{\theta }_{0}}}{A}=-Kt\]

\[\Rightarrow \] \[\theta -{{\theta }_{0}}=A{{e}^{-kt}}\]

which indicates temperature decreases exponentially with increasing time.

(3) Curve between the rate of cooling (R) and body temperature \[(\theta )\].

\[R=K(\theta -{{\theta }_{0}})=K\theta -K{{\theta }_{0}}\] This is a straight line intercept R-axis at \[-K{{\theta }_{0}}\]

(4) Curve between rate of cooling (R) and temperature difference between  body \[(\theta )\] and surrounding \[({{\theta }_{0}})\]

 \[R\propto (\theta -{{\theta }_{0}})\]. This is a straight line passing through origin.