(a) The increment in the internal energy is ΔU=∫V1^v1((∂U)/(∂V))T d V But from second law ((∂U)/

step 1 Hi everyone, increase in internal energy is defined as V1 2v2, D .Dade U upon D .B at constant t

(a) The increment in the internal energy is...

(a) The increment in the internal energy is $$ \Delta U=\int_{V_{1}}^{v_{1}}\left(\frac{\partial U}{\partial V}\right)_{T} d V $$ But from second law $$ \left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial S}{\partial V}\right)_{T}-p=T\left(\frac{\partial p}{\partial T}\right)_{V}-p $$ On the other hand $\quad p=\frac{R T}{V-b}-\frac{a}{V^{2}}$ or, $$ T\left(\frac{\partial p}{\partial T}\right)_{V}=\frac{R T}{V-b} \text { and }\left(\frac{\partial U}{\partial V}\right)_{t}=\frac{a}{V^{2}} $$ So, $\quad \Delta U=a\left(\frac{1}{V_{1}}-\frac{1}{V_{2}}\right)$ (b) From the first law $$ Q=A+\Delta U=R T \ln \frac{V_{2}-b}{V_{1}-b} $$

(a) The increment in the internal energy is
$$
\Delta U=\int_{V_{1}}^{v_{1}}\left(\frac{\partial U}{\partial V}\right)_{T} d V
$$
But from second law
$$
\left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial S}{\partial V}\right)_{T}-p=T\left(\frac{\partial p}{\partial T}\right)_{V}-p
$$
On the other hand $\quad p=\frac{R T}{V-b}-\frac{a}{V^{2}}$
or,
$$
T\left(\frac{\partial p}{\partial T}\right)_{V}=\frac{R T}{V-b} \text { and }\left(\frac{\partial U}{\partial V}\right)_{t}=\frac{a}{V^{2}}
$$
So, $\quad \Delta U=a\left(\frac{1}{V_{1}}-\frac{1}{V_{2}}\right)$
(b) From the first law
$$
Q=A+\Delta U=R T \ln \frac{V_{2}-b}{V_{1}-b}
$$

Key Concepts

Equation of State

An equation of state is a mathematical relationship that links state variables, such as pressure, volume, and temperature, for a given substance. This relationship is crucial for characterizing the thermodynamic behavior of a system and is utilized to predict how the system will respond to changes in these variables.

Thermodynamic Partial Derivatives

Thermodynamic partial derivatives describe how a thermodynamic property changes with respect to one variable while keeping another variable constant. These derivatives are critical for understanding the interdependencies among state variables and are often used to relate properties such as internal energy, entropy, and pressure in a precise mathematical framework.

Maxwell Relations

Maxwell relations are a set of equations derived from the equality of mixed second partial derivatives of thermodynamic potentials. These relations connect different partial derivatives of state functions, such as pressure, volume, temperature, and entropy, and are instrumental in transforming difficult-to-measure quantities into more accessible ones.

First Law of Thermodynamics

The first law of thermodynamics is the principle of energy conservation, stating that the change in internal energy of a system is equal to the heat supplied to the system minus the work done by the system. This law provides a quantitative relationship linking heat, work, and the internal energy of a system, thereby forming the basis for analyzing energy transfer processes.

Second Law of Thermodynamics

The second law of thermodynamics introduces the concept of entropy, explaining that heat transfer processes occur in a direction that increases the overall entropy of the universe. It also provides the basis for understanding relationships between temperature, entropy, and other thermodynamic properties, which can be utilized to derive essential thermodynamic identities and Maxwell relations.

Internal Energy

Internal energy is a fundamental thermodynamic property representing the total energy contained within a system, including both kinetic and potential energies at the microscopic level. It changes as heat is added or work is done by or on the system, and its variation can be expressed differentially in terms of other thermodynamic variables.

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