Question

$\mathbf{300}\mathbf{g}$of water whose temperature is${\mathbf{\text{25}}}^{\mathbf{\text{o}}}\mathbf{\text{C}}$are added to a thin glass containing 800 g of water at${\mathbf{\text{20}}}^{\mathbf{\text{o}}}\mathbf{\text{C}}$(about room temperature). What is the final temperature of the water? What simplifying assumptions did you have to make in order to determine your approximate result?

Short answer

The final temperature of the water after adding it to thin glass is ${\text{21.36}}^{\text{°}}\text{C}$.

The simplifying assumption which helped in determining the approximate result is the heat transfer through the glass is negligible.

Step-by-step solution

Define the energy Principle

The free energy principle defines how non-equilibrium steady-states are maintained in living and non-living systems by restricting their states to a minimal number. It establishes that systems minimise a free energy function of their internal states, implying hidden states in the environment.

Calculation of final temperature of water

Using the Energy Principle, determine the water's ultimate temperature. Subscriptis used to indicate the$300\mathrm{g}$ of water and subscript $2$to designate the water already in the glass ( $800\mathrm{g}$).

Take the sum of both the energies${\mathrm{\Delta E}}_1\text{and}\mathrm{\Delta }{\text{E}}_2$

$\begin{array}{l}{\mathrm{\Delta E}}_1+{\mathrm{\Delta E}}_2=0\\ {\mathrm{m}}_1{\mathrm{C}}_{\mathrm{H}20}({\mathrm{T}}_{\mathrm{f}}-{\mathrm{T}}_{1,\mathrm{i}})+{\mathrm{m}}_2{\mathrm{C}}_{\mathrm{H}20}({\mathrm{T}}_{\mathrm{f}}-{\mathrm{T}}_{2,\mathrm{i}})=0\end{array}$

Here Solve the equation for ${\mathrm{T}}_{\mathrm{f}}$.

$\begin{array}{c}{\mathrm{T}}_{\mathrm{f}}=\frac{{\mathrm{m}}_1{\mathrm{T}}_{1,\mathrm{i}}+{\mathrm{m}}_2{\mathrm{T}}_{2,\mathrm{i}}}{{\mathrm{m}}_1+{\mathrm{m}}_2}\\ =\frac{(300\mathrm{g})\left({25}^{°}\mathrm{C}\right)+(800\mathrm{g})\left({20}^{°}\mathrm{C}\right)}{300\mathrm{g}+800\mathrm{g}}\\ ={21.36}^{°}\mathrm{C}\end{array}$

Assumptions made to approximate result

The following assumptions had to be made in order to calculate the final temperature:

• As the glass is so thin, heat transmission between water and the glass is minimal.
• No heat is sent to the environment.
• Enough time has passed for the ultimate temperature to be attained.

Therefore, the final temperature of the water is$21.36°\mathrm{C}$ and the assumption made is heat transfer only occurs between the two masses of water.