Specific Heats MCQ Quiz - Objective Question with Answer for Specific Heats - Download Free PDF

Explanation:

For an ideal gas, the relationship between the specific heat capacities at constant pressure (Cp) and constant volume (Cv) is given by the equation:

Cp - Cv = R

where R is the universal gas constant. This equation indicates that Cp is always greater than Cv because R is a positive constant.

∴ Cp > Cv is the correct answer.

Concept:

• Heat required to raise the temperature of ice from -15°C to 0°C: Q₁ = m × c₁ × ΔT
• Heat required to melt ice at 0°C: Q₂ = m × L
• Heat required to raise the temperature of resulting water from 0°C to 10°C: Q₃ = m × c₂ × ΔT
• Total heat required: Q = Q₁ + Q₂ + Q₃

 

Calculation:

Mass of ice, m = 700 g = 0.7 kg

Initial temperature of ice, T₁ = -15°C

Final temperature of water, T₂ = 10°C

Specific heat of ice, c₁ = 2220 J/kg·K

Specific heat of water, c₂ = 4187 J/kg·K

Latent heat of fusion of ice, L = 333 kJ/kg

⇒ Heat to raise ice temperature:

Q₁ = 0.7 × 2220 × (0 - (-15))

⇒ Q₁ = 0.7 × 2220 × 15

⇒ Q₁ = 23.31 kJ

 

⇒ Heat to melt ice:

Q₂ = 0.7 × 333

⇒ Q₂ = 233.1 kJ

 

⇒ Heat to raise water temperature:

Q₃ = 0.7 × 4187 × (10 - 0)

⇒ Q₃ = 0.7 × 4187 × 10

⇒ Q₃ = 29.3 kJ

 

⇒ Total heat required:

Q = Q₁ + Q₂ + Q₃

⇒ Q = 23.31 + 233.1 + 29.3

⇒ Q ≈ 286 kJ

∴ The total heat required is approximately 286 kJ.

Calculation:
We use the first law of thermodynamics:

ΔU = nC_vΔT

For a process at constant pressure, we can relate the change in temperature (ΔT) to the change in volume (ΔV) using the ideal gas law:

PΔV = nRΔT

Given that the volume changes from V to 3V, ΔV = 2V. Substituting this into the equation:

P × 2V = nRΔT

Now, using the relation between specific heats:

γ = C_p / C_v

We also know:

C_p - C_v = R

Solving for C_v gives:

C_v = R / (γ - 1)

Thus, the change in internal energy can be expressed as:

ΔU = 2(R / (γ - 1))ΔT = (2P × V) / (γ - 1)

The change in internal energy when the volume changes from V to 3V at constant pressure is:

ΔU = (2PV) / (γ - 1)

The question involves the concept of phase change in thermodynamics, specifically the latent heat of vaporization. When a substance changes its phase from liquid to gas at its saturation temperature, the heat absorbed during this process is termed as the latent heat of vaporization. This heat does not cause a change in temperature but is used to overcome the intermolecular forces holding the liquid molecules together, allowing them to move apart and form vapor. In the case of water, when 1 kg of water at its boiling point (100°C at standard atmospheric pressure) absorbs heat, it changes into 1 kg of steam without any temperature change. This process is critical in many industrial and natural processes, such as in steam engines, power plants, and the water cycle. The correct option in the given question is option 2, "Latent heat of vaporisation."

Calculation: We are given the expressions for the specific heats at constant volume (\(C_V\)) and at constant pressure (\(C_P\)) in terms of the gas constant (\(R\)) and the specific heat ratio (\(r\)).

We need to identify the correct expression among the given options. Let us examine each option:

1) \(\rm C_{V}=\frac{R}{r+1}\)

This expression is not correct because \(C_V\) is typically expressed in terms of \(\frac{R}{r-1}\) for ideal gases.

2) \(\mathrm{C}_{\mathrm{P}}=\frac{\mathrm{rR}}{\mathrm{r}-1}\)

This expression is correct. For an ideal gas, \(C_P\) can be derived from the relation \(C_P = C_V + R\), and using the specific heat ratio \(r = \frac{C_P}{C_V}\), we can manipulate the expressions to get \(C_P = \frac{rR}{r-1}\).

3) \(\rm C_{V}=\frac{r R}{r+1}\)

This expression is not correct as \(C_V\) does not take this form.

4) \(\mathrm{C}_{\mathrm{P}}=\frac{\mathrm{R}}{\mathrm{r}+1}\)

This expression is not correct for \(C_P\) in terms of \(R\) and \(r\).

Final Answer: The correct expression is option 2: \(\mathrm{C}_{\mathrm{P}}=\frac{\mathrm{rR}}{\mathrm{r}-1}\).

Concept:

The change in internal energy is given by, \(Δ U = m{c_v}Δ T\)

Calculation:

Given:

V₂ = 2V; V₁ = V, and P₁ = P₂ = P

\(Δ U = m\frac{R}{{\gamma - 1}}\left( {{T_2} - {T_1}} \right)\)

where \({C_v} = \frac{R}{{\gamma - 1}}\) and ΔT = T₂ – T₁

\(Δ U = \frac{1}{{\gamma - 1}}\left( {mR{T_2} - mR{T_1}} \right)\)

As we know from the ideal gas equation PV = mRT.

\(Δ U = \frac{1}{{\gamma - 1}}\left( {{P_2}{V_2} - {P_1}{V_1}} \right)\)

\(Δ U = \frac{P}{{\gamma - 1}}\left( {{}{2V} - {}{V}} \right)\)

∴ we get, \(Δ U = \frac{{PV}}{{\gamma - 1}}\).

Mistake Points

In the Isobaric process, pressure is constant throughout the process.

ΔW = P₂V₂ - P₁V₁ = mR(T₂ -T₁)

ΔQ = mC_p(T2 -T1)

ΔU = mC_v(T2 -T1)

Hence for isobaric process also, the change in internal energy is given by ΔU = mCv(T2 -T1)

Concept:

When Q joule heat is added to a body whose mass is m, the temperature rises from T₁ to T₂.

It is given by Q = mc(T₂ - T₁) where c = specific heat of the body.

Calculation:

Given:

m = 2 kg, Q = 500 kJ, T₂ = 200 °C, T₁ = 100 °C   

∵ Q = mc(T₂ – T₁)

⇒ 500 = 2 × c × (200 – 100)

⇒ c = 2.5 kJ/kg°K

Important Points

When difference of temperature is needed do not convert °C into °K.

CONCEPT:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{\Delta Q}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The ratio of the two principal specific heat is represented by γ.

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

• The value of γ depends on the atomicity of the gas.

CALCULATION:

Given - C_v = 8 cal/mol-K and R = 4 cal/mol-K

• For an ideal gas,

⇒ Cp - Cv = R

⇒ C_p = C_v + R = (8 + 4) = 12 cal/mol-K

• The ratio of the two principal specific heat is represented by γ.

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

\(\Rightarrow \gamma = \frac{{12}}{8} = 1.5\)

Concept:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant volume.

\({C_v} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;volume}}\)Additional Information

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1 °C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Where f = degree of freedom

For monoatomic gas f = 3

For Diatomic gas f = 5

Concept:

A perfect gas is also an Ideal gas, which follows the Ideal gas equation of states i.e. PV = mRT all temperature.

where, P = pressure of gas, V = volume occupied, m = mass of a gas, R = universal gas constant. 

The universal gas constant (R) is the difference between specific heat constants for constant pressure (C_p) and constant volume (C_v)

i.e.R = C_p - C_v

A real gas behaves as an Ideal gas at low pressure and very high temperature. Air is a perfect gas.

Gases that obey the gas laws (Charles law, Boyles law, and Universal Gas Law) are called ideal gases.

Boyle’s, Charles’, and Gay Lussac's Laws describe the basic behavior of fluids with respect to volume, pressure, and temperature.

Gay Lussac’s Law	It states that at constant volume, the pressure of a fixed amount of a gas varies directly with temperature. P ∝ T \(\frac{P}{T} = Const\)
Boyle's Law	For a fixed mass of gas at a constant temperature, the volume is inversely proportional to the pressure. \(P\propto \frac{1}{V}\) PV = constant (If the temperature remains constant, the product of pressure and volume of a given mass of a gas is constant.)
Charles' Law	For a fixed mass of gas at constant pressure, the volume is directly proportional to the Kelvin temperature. \(V\propto T \ or, \ \frac{V}{T} = Const\)

The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas:

\(\frac{P_{1}V_{1}}{T_{1}} = \frac{P_{2}V_{2}}{T_{2}}\)

\({C_P} - {C_V} = \frac{R}{J}\) is valid for a perfect gas only.

Explanation:

Specific heat capacity (c) of a substance is defined as the amount of heat (ΔQ) per unit mass of the substance that is required to raise the temperature (ΔT) by 1°C.

\(c=\frac{1}{m} \frac{ΔQ}{ΔT}\)

where m = mass of kg, ΔQ = heat added or removed in J, and ΔT = heat increased or decreased in Kelvin (K).

The unit of specific heat capacity is Jkg^-1K^-1.

Concept:

Heat absorbed = Latent Heat + sensible heat

Latent heat of fusion(LH)  = m × Latent heat of fusion per kg(LH)

Sensible heat(SH) = m × specific heat × ΔT

where m = mass, ΔT = temperature change

Calculation:

Given:

m = 500 gram, Specific heat of Ice = 2.2 kJ/kg K, Specific heat of water = 4.2 kJ/kg K, Latent heat of fusion of ice = 300 kJ/kg)

Total heat absorbed = SH of ice + LH of fusion + SH of water

Total Heat absorbed = (0.5 × 2.2 × 10) + (0.5 × 300) + (0.5 × 4.2 × 20)

Total Heat absorbed = 11 + 150 + 42 = 203 kJ

Explanation:

Types of gas	cv	cp	\(\gamma =\frac {c_p}{c_v}\)
Monoatomic	(3/2)R	(5/2)R	1.67
Diatomic	(5/2)R	(7/2)R	1.4
Tri - atomic	(7/2)R	(9/2)R	1.28

 

∴ γ = 1.4 for air, which is predominantly a diatomic gas.

Concept:

• Latent heat: It is the amount of heat needed to change the state of a unit mass of any substance without changing its temperature.
• i.e., we can say that heat needed to supply for changing its state will be directly proportional to the mass m of the substance.

Q ∝ m

i.e.,Q = mL

• The SI unit of Latent heat is J/Kg and the CGS unit is Cal/g respectively.
• Latent heat is the hidden heat that is required to change the state of a substance. There is no change in temperature because all the heat is used to overcome the attractive forces between the molecules.

Specific heat:

• It is the amount of heat required to raise the temperature of a gram of a substance by one degree.
• Its unit is calories or joule per gram per degree Celcius: J/g/° 

Explanation:

Latent heat of fusion:

• When the melting point of a substance is reached, the temperature of a substance does not change even though we continue to heat the substance. This additional heat gets used up in changing the state by overcoming forces of attraction between the particles. This is called as latent heat of fusion.
• It is defined as the amount of heat energy that is required to change 1 kg of a solid into liquid at atmospheric pressure at its melting point.
• The latent heat of the ice is 80 Cal/g and 3.34 × 105 J/kg

Latent heat of sublimation:

• It is the heat required per unit mass to change the state of a substance from solid to gas directly at a constant temperature.
• The substances which undergo sublimation are Naphthalene, camphor etc.

Latent heat of evaporation:

• The heat absorbed per unit mass of a given material at its boiling point which completely converts the material to a gas at the same temperature is called Latent heat of evaporation.

​Hence, the amount of heat required for converting one kilogram of a solid completely into liquid is called Latent heat of fusion. 

Concept:

• The molar specific heat capacity of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1°C at the constant volume.

\({C_v} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;volume}}\)

• The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 mol of the gas by 1°C at the constant pressure.

\({C_p} = {\left( {\frac{{\Delta Q}}{{n\Delta T}}} \right)_{constant\;pressure}}\)

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Where f = degree of freedom

EXPLANATION:

• The relation between the ratio of Cp and Cv with a degree of freedom is given by

\(\Rightarrow \gamma = \frac{{{C_p}}}{{{C_v}}} = 1 + \frac{2}{f}\)

Monoatomic gas has 3 degrees of freedom

\(\Rightarrow \gamma = 1 + \frac{2}{3} = \frac{3 +2}{3} = \frac{5}{3} =1.67\)