Thermodynamics Concept Page - 6

Definition

Perpetual Motion Machine

The perpetual motion machine is a machine where there is only one heat reservoir involved, which is being spontaneously cooled without involving a transfer of heat to a cooler reservoir. This conversion of heat into useful work, without any side effect, is impossible, according to the second law of thermodynamics.

Definition

Vander Waal's equation of state

• The constant a provides a correction for the intermolecular forces. 

• The constant b is a correction for finite molecular size. Its value is the volume of one mole of the atoms or molecules.

1. There is interaction forces which act between the molecules of real gases. This was first considered by van der Wall. 
2. In the equation of state of the perfect gas, it is assumed that gas molecules do not interact with one another. But this isnt true always.
3. Hence, the ideal gas equation was modified considering those interactions as below 

Definition

Path variable

Path variables depend on the path taken to reach one state from another to define a thermodynamic process.Examples of path variables include work,heat etc.

Definition

Mathematical Condition for state variables

State functions depend on three things: the property, the initial value, and the final value. In other words, integrals illustrate how state functions depend only on the final and initial value and not on the object's history or the path taken to get from the initial to the final value.
For example, enthalpy H$H$ is a state function, so:
∫t1t0H(t)dt=H(t1)−H(t0)=ΔH${\int }_{t_0}^{t_1}H(t)dt=H(t_1)-H(t_0)=\mathrm{\Delta }H$
where the subsripts 1 and 0 represent final and initial states respectively.On the other hand, multiple integrals and multiple limits of integration are required to take the integral of a path function.

Definition

Molar Specific Heat Capacity

The specific capacity is the property of the substance which determines the change in the temperature of the substance (undergoing no phase change) when a given quantity of heat is absorbed (or rejected) by it. It is defined as the amount of heat per unit mass absorbed or rejected by the substance to change its temperature by one unit. It depends on the nature of the substance and its temperature. The molar specific heat capacity is the specific heat capacity per mole of the substance. If the amount of substance is specified in terms of mole μ$\mu$, instead of mass m$m$ in kg, we can define heat capacity per mole of the substance by
C=Sμ=1μδQδT$C={\displaystyle \frac{S}{\mu }}={\displaystyle \frac{1}{\mu }}\delta Q\delta T$
where C is known as molar specific heat capacity of the substance. The SI unit of molar specific heat capacity is Jmol−1K−1$Jmo{l}^{-1}{K}^{-1}$.

Formula

Specific heat capacity of polyatomic gas

In general a polyatomic molecule has 3 translational, 3 rotational degrees of freedom and a certain number (f) of vibrational modes.
According to the law of equipartition of energy, it is easily seen that one mole of such a gas has
U=(3KBT2+3KBT2+fKBT)NA$U=({\displaystyle \frac{3K_{B}T}{2}}+{\displaystyle \frac{3K_{B}T}{2}}+f{K}_{B}T)N_A$ 
or Cv=(3+f)R$C_v=(3+f)R$, Cv=(4+f)R$C_v=(4+f)R$

Formula

Specific heat capacity of diatomic gas

The molecules of a monatomic gas have 5 degrees of freedom, 3 translational and 2 rotational.
The average energy of a molecule at temperature T is 5KBT2${\displaystyle \frac{5K_{B}T}{2}}$.
The total internal energy of a mole is:  5KBT2×NA${\displaystyle \frac{5K_{B}T}{2}}\times N_A$.
The molar specific heat at constant volume Cv$C_v$ is 
For an ideal gas,
Cv(monatomicgas)=dUdT=5RT2$C_v(monatomicgas)={\displaystyle \frac{dU}{dT}}={\displaystyle \frac{5RT}{2}}$
FOr an ideal gas, Cp−Cv=R$C_p-C_v=R$
where Cp$C_p$ is molar specific heat at constant pressure. 
Thus, Cp=7R2$C_p={\displaystyle \frac{7R}{2}}$

Formula

Specific heat capacity of monatomic gas

The molecules of a monatomic gas have 3 degrees of freedom.
The average energy of a molecule at temperature T is 3KBT2${\displaystyle \frac{3K_{B}T}{2}}$.
The total internal energy of a mole is:  3KBT2×NA${\displaystyle \frac{3K_{B}T}{2}}\times N_A$.
The molar specific heat at constant volume Cv$C_v$ is 
For an ideal gas,
Cv(monatomicgas)=dUdT=3RT2$C_v(monatomicgas)={\displaystyle \frac{dU}{dT}}={\displaystyle \frac{3RT}{2}}$
FOr an ideal gas, Cp−Cv=R$C_p-C_v=R$
where Cp$C_p$ is molar specific heat at constant pressure.
Thus, Cp=5R2$C_p={\displaystyle \frac{5R}{2}}$

Example

Measuring specific heat capacity of a solid in calorimeter

Measurement of specific heat capacity of solid is done in a calorimeter. 
c=[(m2−m1)cw+m1cc](t−t1)m(t2−t)$c={\displaystyle \frac{[(m_2-m_1)c_w+m_1{c}_c](t-t_1)}{m(t_2-t)}}$
where 
cw:$c_w:$ specific heat capacity of water
cc:$c_c:$ specific heat capacity of calorimeter
m2:$m_2:$ mass of calorimeter + water
m1:$m_1:$ mass of calorimeter
m:$m:$ mass of solid
t2:$t_2:$ temperature of solid
t1:$t_1:$ initial temperature of water
t:$t:$ temperature of mixture

Example

Measuring specific heat capacity of a liquid in calorimeter

Measurement of specific heat capacity of liquid is done in a calorimeter. 
cL=mc(t2−t)−m1cc(t−t1)(m2−m1)(t−t1)$c_L={\displaystyle \frac{mc(t_2-t)-m_1{c}_c(t-t_1)}{(m_2-m_1)(t-t_1)}}$
where 
c:$c:$ specific heat capacity of known solid
cc:$c_c:$ specific heat capacity of calorimeter
m2:$m_2:$ mass of calorimeter + liquid
m1:$m_1:$ mass of calorimeter
m:$m:$ mass of solid
t2:$t_2:$ temperature of solid
t1:$t_1:$ initial temperature of liquid
t:$t:$ temperature of mixture

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