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Practice each question, then open the answer or worked solution.
Degrees of Freedom: The number of degrees of freedom of a dynamical system is defined as the total number of co-ordinates or independent quantities required to describe completely the position and configuration of the system.
Solution coming soon.
Law of Equipartition of Energy: According to this law, for any dynamical system in thermal equilibrium, the total energy is distributed equally amongst all the degrees of freedom, and the energy associated with each molecule per degree of freedom is $\frac{1}{2}k_{B}T$ where kB is Boltzman constant, and T is temperature of the system.
Solution coming soon.
Mean Free Path: The path traversed by a molecule between two successive collisions with another molecule is called the mean free path
$1\vec{} = \frac{\text{Total distance travelled by a molecule}}{\text{No. of collisions it makes with other molecules}}$
Thermodynamics
The Zeroth Law: The zeroth law of thermodynamics states that ‘two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other’.
Internal energy: Internal energy of a system is the sum of kinetic energies and potential energies of the molecular constituents of the system.
First law of thermodynamics: This is the general law of conservation of energy applied to any system in which energy transfer from or to the surroundings (through heat and work) is taken into account. It states that
$\Delta Q=\Delta U+\Delta W$
Where, $\Delta Q$ is the heat supplied to the system.
$\Delta W$ is the work done by the system and $\Delta U$ is the change in internal energy of the system.
Specific heat capacity: Specific heat capacity of a substance is defined by $s=\frac{1}{m}\frac{\Delta Q}{\Delta T}$
Where m is the mass of the substance and $\Delta Q$ is the heat required to change its temperature by $\Delta T$.
Molar specific heat capacity: Molar specific heat capacity of a substance is defined by $C=\frac{1}{\mu }\frac{\Delta Q}{\Delta T}$
Where, $\mu$ is the number of moles of the substance.
Mayer’s Law: For an ideal gas, the molar specific heat capacities at constant pressure (Cp) and at constant volume (Cv) satisfy the relation.
Cp- Cv = R
where R is the universal gas constant.
State Variables: Equilibrium states of a thermodynamic system are described by state variables.
The value of a state variable depends only on the particular state, not on the path used to arrive at thestate. Examples of state variables are pressure (p), volume (V), temperature (T), and mass (m).
Isothermal Process: In an isothermal expansion of an ideal gas from volume V1 to V2 at temperature T the heat absorbed (Q) equals the work done (W) by the gas, each given by
$Q=W=\mu RT\left ( \frac{V_{2}}{V_{1}} \right )$
Adiabatic Process: In an adiabatic process of an ideal gas $PV^{\gamma }=Constant$
Where $\gamma=\frac{C_{p}}{C_{v}}$
Work done by an ideal gas in an adiabatic change of state from(P1,V1,T1) to (P1,V2,T2) is
$W=\frac{\mu RT\left ( T_{1} +T_{2}\right )}{\gamma -1}$
Second law of thermodynamics: Second law of thermodynamics disallows some processes consistent with the First Law of Thermodynamics.
Kelvin -Planck statement: No process is possible whose sole result is the absorption of heat from a reservoir and complete conversion of the heat into work.
Clausius statement: No process is possible whose sole result the transfer of heat from a colder object to a hotter object.
The second Law implies that no heat engine can have efficiency η equal to 1 or no refrigerator can have co-efficient of performance α equal to infinity.
Reversible Process: A process is reversible if it can be reversed such that both the system and the surroundings return to their original states, with no other change anywhere else in the universe. Spontaneous processes of nature are irreversible. The idealized reversible process is a quasi-static process with no dissipative factors such as friction, viscosity, etc.
Carnot engine: Carnot engine is a reversible engine operating between two temperatures T1 (source) and T2 (sink). The Carnot cycle consists of two isothermal processes connected by two adiabatic processes. The efficiency of a Carnot engine is given by
$\eta = 1 - \frac{T_{2}}{T_{1}}$
No engine operating between two temperatures can have efficiency greater than that of the Carnot engine.
Solution coming soon.