Solve the Problem $\mathrm { dU } = \mathrm { T } \mathrm { dS } - \mathrm { P } \mathrm { dV }$

question 98

Essay

Solve the problem.
-In thermodynamics, the differential form of the internal energy of a system is $\mathrm { dU } = \mathrm { T } \mathrm { dS } - \mathrm { P } \mathrm { dV }$ , where $\mathrm { U }$ is the internal energy, $\mathrm { T }$ is the temperature, $\mathrm { S }$ is the entropy, $\mathrm { P }$ is the pressure, and $\mathrm { V }$ is the volume of the system. The First Law of Thermodynamics asserts that dU is an exact differential. Using this information, justify the thermodynamic relation $\frac { \partial \mathrm { T } } { \partial \mathrm { V } } = - \frac { \partial \mathrm { P } } { \partial \mathrm { S } }$ .

---

Definitions:

Investment

The allocation of resources, typically money, in order to gain profitable returns, as interest, income, or appreciation in value.

Federal Funds Target Rate

The interest rate at which banks and credit unions lend reserve balances to other depository institutions overnight, set by the Federal Reserve to influence economic conditions.

Government Bonds

Debt securities issued by a government to finance its expenditures, typically featuring a fixed interest rate and maturity date.

Interest Rates

Represents the charge or payment for the use of money, often expressed as an annual percentage of the loan or investment amount.