ΔS=-(m q1)/(T2)-m c ln(T2)/(T1)+(M qi c e)/(T1) where M qi c e=m(q2+c(T2-T1))

step 1 Hello students in this question we have molten lead of mass m equals to 5 gram at temperature t2

ΔS=-(m q1)/(T2)-m c ln(T2)/(T1)+(M qi c e)/(T1) ...

$$ \begin{aligned} &\Delta S=-\frac{m q_{1}}{T_{2}}-m c \ln \frac{T_{2}}{T_{1}}+\frac{M q_{i c e}}{T_{1}} \\ &\text { where } \quad M q_{i c e}=m\left(q_{2}+c\left(T_{2}-T_{1}\right)\right) \\ &=m q_{2}\left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right)+m c\left(\frac{T_{2}}{T_{1}}-1-\frac{T_{2}}{T_{1}}\right) \\ &=0.2245+0.2564 \sim 0.48 \mathrm{~J} / \mathrm{K} \end{aligned} $$

$$
\begin{aligned}
&\Delta S=-\frac{m q_{1}}{T_{2}}-m c \ln \frac{T_{2}}{T_{1}}+\frac{M q_{i c e}}{T_{1}} \\
&\text { where } \quad M q_{i c e}=m\left(q_{2}+c\left(T_{2}-T_{1}\right)\right) \\
&=m q_{2}\left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right)+m c\left(\frac{T_{2}}{T_{1}}-1-\frac{T_{2}}{T_{1}}\right) \\
&=0.2245+0.2564 \sim 0.48 \mathrm{~J} / \mathrm{K}
\end{aligned}
$$

Key Concepts

Entropy

Entropy is a fundamental concept in thermodynamics that represents the degree of disorder or randomness within a system. It quantifies the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state and is directly related to the amount of heat transferred at a specific temperature. Entropy change, often calculated using the relation ?S = ?(dQ/T) for reversible processes, is central to understanding the direction of spontaneous processes and the second law of thermodynamics.

Heat Transfer

Heat transfer in thermodynamics refers to the movement of thermal energy from one system to another or within a system due to a temperature gradient. It plays a critical role in determining the entropy change of a system since the amount and manner of heat transferred (whether at constant or varying temperatures) will dictate the system’s thermal evolution and its approach to equilibrium.

Specific Heat Capacity

Specific heat capacity is a property of a material that defines the amount of heat required to raise the temperature of a unit mass of the substance by one degree. In thermodynamic processes, particularly those involving temperature changes, specific heat capacity is used to relate the heat added or removed (dQ) to the corresponding change in temperature (dT), which in turn influences the entropy change through terms like m c ln(T2/T1).

Latent Heat

Latent heat is the energy absorbed or released by a substance during a phase transition, such as melting or freezing, at a constant temperature. Unlike sensible heat, which changes the temperature of a substance, latent heat leads to a change in the state of the material without changing its temperature, and it must be accounted for when calculating the overall energy and entropy changes during phase transitions.

Integration of Entropy over Temperature

The integration of entropy with respect to temperature involves expressions where the temperature is not constant over the process. When a system undergoes a change in temperature, the change in entropy is calculated by integrating the heat capacity over the temperature range, often leading to logarithmic relationships. This mathematical treatment is key to understanding how entropy changes due to varying thermal conditions during a process.

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