Suppose 1.00 kg of water at 10.0^∘ C is mixed with 1.00 kg of water at 30.0^∘ C at constant pres

step 1 In this question we have 1kG of water at 10 degrees Celsius mixing with 10 degree of water at 30

Suppose 1.00 kg of water at 10.0^∘ C is mixed with 1.00 kg...

Suppose 1.00 $\mathrm{kg}$ of water at $10.0^{\circ} \mathrm{C}$ is mixed with 1.00 $\mathrm{kg}$ of water at $30.0^{\circ} \mathrm{C}$ at constant pressure. When the mixture has reached equilibrium, (a) what is the final temperature? (b) Take $c_{P}=4.19 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}$ for water and show that the entropy of the system increases by $$\Delta S=4.19 \ln \left[\left(\frac{293}{283}\right)\left(\frac{293}{303}\right)\right] \mathrm{kJ} / \mathrm{K}$$ (c) Verify numerically that $\Delta S>0 .$ (d) Is the mixing an irreversible process?

Suppose 1.00 $\mathrm{kg}$ of water at $10.0^{\circ} \mathrm{C}$ is mixed with 1.00 $\mathrm{kg}$ of water at $30.0^{\circ} \mathrm{C}$ at constant pressure. When the mixture has reached equilibrium, (a) what is the final temperature? (b) Take $c_{P}=4.19 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}$ for water and show that the entropy of the system increases by
$$\Delta S=4.19 \ln \left[\left(\frac{293}{283}\right)\left(\frac{293}{303}\right)\right] \mathrm{kJ} / \mathrm{K}$$
(c) Verify numerically that $\Delta S>0 .$ (d) Is the mixing an irreversible process?

Key Concepts

Thermal Equilibrium

This concept refers to the state in which two or more objects or substances, when in thermal contact, reach a common temperature so that there is no net heat transfer between them. In the context of mixing liquids at different temperatures, thermal equilibrium is achieved when the heat lost by the warmer substance equals the heat gained by the cooler substance.

Conservation of Energy

This principle states that the total energy in an isolated system remains constant. When mixing substances of different temperatures, the heat lost by the hotter substance is equal to the heat gained by the cooler substance. This balance is used to determine the equilibrium temperature, under the assumption that no heat is lost to the environment.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree. It is a key property in calculating energy exchange during temperature changes. In problems involving heat transfer, specific heat is used to quantify the energy change associated with a change in temperature.

Entropy Change

Entropy is a thermodynamic quantity that measures the degree of disorder or randomness in a system. When two substances mix or a system undergoes a temperature change, the entropy typically increases if the process is irreversible. Calculations of entropy change integrate the heat transfer over the temperature at which the transfer occurs, reflecting the intrinsic irreversibility of the process.

Irreversible Processes

An irreversible process is one in which the system and its surroundings cannot be returned to their original states without additional changes in the universe. When mixing substances at different initial temperatures, the resulting entropy increase indicates that the process is irreversible, even if the final state is a stable thermal equilibrium.

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