Option 2 : Helium

In case of gas, we get two specific heats. They are specific heat at constant pressure and specific heat at constant volume. The ratio of these two specific heats is given by γ. In case of object, the degrees of freedom mean the number of independent variables to understand the object.

For ideal gas (obeys PV = nRT law) the relation between specific heat ratio and degrees of freedom is γ = (f + 2)/f.

For example, a monatomic gas has three DOF, so we get γ = (3 + 2)/3 = 5/3.

The specific heats of an ideal gas depends on its

Option 4 : Molecular weight and structure

__Explanation:__

**The specific heat of ideal gas depends on molecular weight and structure.**

**Specific heat or specific heat capacity** of a body is the amount of heat required for a unit mass of the body to raise the temperature by 1 degree Celsius.

- It is denoted by
**“c” or “s”.** - It is different for each substance.
- Its unit is cal/g-°C in CGS and J/kg-K in the SI unit.
- 1 calorie = 4.186 Joule.
- As the specific heat of a gas is the heat required for the unit mass of the body to raise the temperature by unit degree Celsius, so it is independent of the temperature of the gas.

Option 3 : \(\frac{{b\; + \;1}}{b}\)

**Concept:**

From 1st law of thermodynamics,

δq = du + δw

⇒ δq = du + pdv for a closed system or non-flow process

and for reversible adiabatic process q = 0

⇒ du + pdv = 0 ----(1)

**Calculation:**

**Given:**

u = a + bpv

du = 0 + bp.dv + bv.dp ----(2)

From equation (1) & (2)

bp.dv + bv.dp + pdv = 0

(b + 1)pdv + bv.dp = 0

Dividing both sides by bvp:

\(\frac{{b \ + 1}}{b}\frac{{dv}}{v} + \frac{{dp}}{p} = 0\)

\(\frac{{b \;+ \;1}}{b}\ln V + \ln P = \ln c\)

\(p{V^{\left( {\frac{{b + 1}}{b}} \right)}} = c\)

For reversible adiabatic process:

pVγ = const

Thus,**\(\gamma = \frac{{b \;+\; 1}}{b}\)**

Option 4 : 1 + 2/n

**Explanation:**

Number of degrees of freedom in a polyatomic gas molecule = n

Internal energy of one-gram mole of the gas:

\(\begin{array}{l} U = \frac{n}{2}RT\\ {C_v} = \frac{{dU}}{{dT}} = \frac{n}{2}R \end{array}\)

\({C_P} = {C_v} + R = \left( {\frac{n}{2} + 1} \right)R\)

\(\frac{{{C_P}}}{{{C_v}}} = \gamma = \frac{{\left( {\frac{n}{2} + 1} \right)}}{{\frac{n}{2}}} = \frac{{n + 2}}{n} = 1 + \frac{2}{n}\)

Option 2 : Helium

In case of gas, we get two specific heats. They are specific heat at constant pressure and specific heat at constant volume. The ratio of these two specific heats is given by γ. In case of object, the degrees of freedom mean the number of independent variables to understand the object.

For ideal gas (obeys PV = nRT law) the relation between specific heat ratio and degrees of freedom is γ = (f + 2)/f.

For example, a monatomic gas has three DOF, so we get γ = (3 + 2)/3 = 5/3.

Option 1 : 1.5

__Concept:__

Mayer’s Equation for an ideal gas:

Cp – Cv = R

The ratio of Specific heat is known as an adiabatic index.

\(\gamma = \frac{{{C_p}}}{{{C_v}}}\)

Where, Cp = Specific heat of gas at constant pressure and Cv = Specific heat of gas at constant volume

∴ The ratio of specific heat of a gas at constant pressure and the specific heat of the gas at constant volume is always constant.

**Calculation:**

**Given:**

R = 0.1 kJ/kg.K, C_{v} = 0.2 kJ/kg.K

Therefore, C_{p} - 0.1 = 0.2

C_{p} = 0.3 kJ/kg.K

The ratio of Specific heat is

\(\gamma = \frac{{{0.3}}}{{{0.2}}}=1.5\)

Option 3 : \({C_{pe}} = \frac{{\sum \left( {m{C_p}} \right)}}{{\sum m}}\)

__Concept:__

R_{e} (equivalent characteristic gas constant)

p_{1}v = m_{1}R_{1}T, p_{2}v = m_{2}RT ---- on adding

(p_{1} + p_{2} + …) V = (∑ mR) T ----(1)

If mixture is considered as single gas

p_{t} × v = (∑m) R_{e} T ----(2)

**On Comparing (1) & (2)**

\({R_e} = \frac{{\sum \left( {mR} \right)}}{{\sum m}}\)

As \(U = {U_1} + {U_2} + \ldots \)

\(\left( {\sum m} \right){C_{ve}}\;T = {m_1}{C_{v1}}\;{T_1} + {m_2}\;{C_{v2}}\;T + \ldots \)

\({C_{ve}} = \frac{{\sum m{C_v}}}{{\sum m}}\)

Similarly,

H = H_{1} + H_{2} …

\(\left( {\sum m} \right){C_{pe}}\;T = {m_1}{C_{p1}}\;T + {m_2}\;{C_{p2}}\;T + \ldots \)

\({C_{pe}} = \frac{{\sum m{C_p}}}{{\sum m}}\)

Option 3 : 99.5 kJ/kg

__Concept:__

For an ideal gas, the change in enthalpy is given as

dH = m C_{P} dT

For constant C_{p}, ΔH = mC_{p} ΔT;

For temperature-dependent C_{p}, we have to integrate.

__Calculation:__

Given CP = 0.9 + (2.7 × 10-4)T (kJ/kgK) where T is in Kelvin;

now dH will be

dH = m (0.9 + (2.7 × 10-4)T) dT;

Integrating on both sides now,

\(\begin{array}{l} \smallint dH = \mathop \smallint \limits_{300}^{400} 0.9 + \left( {2.7 \times {{10}^{ - 4}}} \right)TdT\\ \end{array}\)

\(⇒ Δ H = 0.9T + \left( {2.7 \times {{10}^{-4}}} \right)\left. {\frac{{{T^2}}}{2}} \right|_{300}^{400}\)

The specific heats of an ideal gas depends on its

Option 4 : Molecular weight and structure

__Explanation:__

**The specific heat of ideal gas depends on molecular weight and structure.**

**Specific heat or specific heat capacity** of a body is the amount of heat required for a unit mass of the body to raise the temperature by 1 degree Celsius.

- It is denoted by
**“c” or “s”.** - It is different for each substance.
- Its unit is cal/g-°C in CGS and J/kg-K in the SI unit.
- 1 calorie = 4.186 Joule.
- As the specific heat of a gas is the heat required for the unit mass of the body to raise the temperature by unit degree Celsius, so it is independent of the temperature of the gas.

Option 4 : 1 + 2/n

**Concept:**

Number of degrees of freedom in a polyatomic gas molecule = n

The internal energy of a one-gram mole of the gas:

\(\begin{array}{l} U = \frac{n}{2}RT\\ {C_v} = \frac{{dU}}{{dT}} = \frac{n}{2}R \end{array}\)

\({C_P} = {C_v} + R = \left( {\frac{n}{2} + 1} \right)R\)

\(\frac{{{C_P}}}{{{C_v}}} = \gamma = \frac{{\left( {\frac{n}{2} + 1} \right)}}{{\frac{n}{2}}} = \frac{{n + 2}}{n} = 1 + \frac{2}{n}\)