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

Calculation:

\(\rm\Delta Q_{1}=m \times S_{1} \times \Delta T=10^{-3} \times 2100 \times 10=21 \mathrm{~J}\)

\(\rm\Delta \mathrm{Q}_{2}=\mathrm{m} \times \mathrm{L}_{\mathrm{f}}=10^{-3} \times 3.35 \times 10^{5}=335 \mathrm{~J}\)

\(\rm\Delta Q_{3}=m \times S_{w} \times \Delta T=10^{-3} \times 4180 \times 100=418 \mathrm{~J}\)

\(\rm \Delta \mathrm{Q}_{4}=\mathrm{m} \times \mathrm{L}_{\mathrm{v}}=10^{-3} \times 2.25 \times 10^{6}=2250 \mathrm{~J}\)

\(\rm\Delta Q_{5}=\mathrm{m} \times \mathrm{S}_{\mathrm{v}} \times \Delta \mathrm{T}=10^{-3} \times 1920 \times 10=19.2 \mathrm{~J}\)

\(\rm \Delta Q_{\mathrm{net}}=3043.2 \mathrm{~J}\)

Δu = n₁C₁ΔT + n₂C₂ΔT

= (n₁C₁ + n₂C₂)ΔT                    ...(i)

Work done = PΔv

= (n₁ + n₂)RΔT                         ...(ii)

Divide (i) by (ii)

\(\begin{array}{l} \frac{\Delta u}{W}=\frac{\left(n_{1} C_{1}+n_{2} C_{2}\right) \Delta T}{\left(n_{1}+n_{2}\right) R \Delta T} \\ \Delta u=\frac{W}{R}\left(\frac{n_{1} C_{1}+n_{2} C_{2}}{n_{1}+n_{2}}\right) \\ =\frac{66}{R} \frac{\left[\frac{3 R}{2} \times 2+\frac{5 R}{2} \times 1\right]}{2+1} \end{array}\)

= 121 J

Answer : 4

Solution :

[Note : Since mass of the metal is not mentioned in the question, it is not possible to arrive at the answer.]

Concept:

In an adiabatic process, no heat is exchanged with the surroundings. During sudden expansion, the gas does work on its surroundings, causing its internal energy to decrease, leading to a drop in temperature. Since the process is fast, there’s no time for heat transfer, making it adiabatic.

The air has mostly diatomic molecule like O2, N2 etc.

Explanation:

From adiabatic gas relation ,

\(\text{P}_1^{1-\gamma}\text{T}_1^{\gamma} = \text{P}_2^{1-\gamma}\text{T}_2^{\gamma}\)

Given :

\(\text{P}_1=5 \text{ atm} \\ \text{T}_1=47^\circ \text{ C} \\ \text{P}_2=1 \text{ atm}\\ \gamma =\frac{c_p}{c_v}=1.4\)

Thus the final temeprature T2.

After bursting the pressure of tyre will became the atmospheric pressure. 

\(\text{T}_2=\text{T}_1(\frac{\text{P}_1}{\text{P}_2})^{\frac{1-\gamma}{\gamma}}\\ \quad =320\text{ K} \times 5^{-0.285}\\ \quad =202.2 \text{ K}\)

The correct answer lies from 202.2 K to 202.3 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 relation between the ratio of C_p and C_v 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 C_p and C_v with a degree of freedom is given by

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

Triatomic gas has  6 degrees of freedom : 3 translational and 3 rotational

\(\gamma = 1 + \frac{2}{6} = 1 + \frac{1}{3} = \frac{4}{3} = 1.33\)

Concept:

Compressibility is the reciprocal of the bulk modulus of elasticity.

Compressibility (p) = 1/K, and K = bulk modulus of Elasticity

\({\rm{K}} = \frac{{{\rm{Increase\;of\;pressure}}}}{{{\rm{Volumetric\;strain}}}} = \frac{{{\rm{dP}}}}{{\frac{{ - {\rm{dv}}}}{{\rm{v}}}}} = \frac{{ - {\rm{dP}}}}{{{\rm{dv}}}} \times {\rm{V}}\)      ----(i)

For isothermal process:

\(\frac{{\rm{P}}}{{\rm{\rho }}} = {\rm{Constant}} \Rightarrow {\rm{P}} \times {\rm{V}} = {\rm{constant}}\)      ----(ii)

Differentiating equation (ii),

PdV + Vdp = 0

⇒ PdV = -Vdp

\(\Rightarrow {\rm{P}} = \frac{{ - {\rm{VdP}}}}{{{\rm{dV}}}}\)      ----(iii)

From equation (i) & (iii), we have

K = P

For adiabatic condition, \(\frac{{\rm{P}}}{{{{\rm{\rho }}^{\rm{k}}}}} = \) constant, where k = Ratio of specific heats.

Bulk modulus, K = Pk

Concept:

Heat Capacity

• The heat capacity is the amount of energy required to raise the temperature of the given substance by 1 K or 1 °C. 
• The amount of heat required to raise the temperature of 1 kg of a substance by 1 K or 1 °C is called specific heat capacity. 
• If the heat capacity is H, the energy required to raise the temperature by Δ T is E = H. Δ T
• Specific heat capacity is S, then the energy required will be E = m S  Δ T.

Joules Heating

An electric current passing through any device will produce heat. 

The heat is produced by the resistance offered to the electric current by the device. 

It is given by H = I² Rt = VIt = V² /R × t

Electrical energy = Power × time of consumption

Calculation:

The energy produced by the heater is used to raise the temperature of liquid from 25°C to 31°C in 2 minutes.

Let heat capacity is H

Temperature change =  Δ T  = 31 ° C - 25 ° C = 6 ° C = 6 K

Heat required to raise temperature E  = H × 6 K -- (1)

Power = 1000 W

time t = 2 min = 120 s

Eneegy supplied by heater E = P × t = 1000 W × 120 s  -- (2)

Equating (1) and (2)

H × 6 K = 1000 W × 120 s 

⇒ \(H = \frac{1000 \times 120}{6} \ J/^{\circ}C = 2 \times 10^4 J/^{\circ}C\)

So, the correct option is 2 × 104 J/°C

Mistake PointsHere the question is about heat capacity not specific heat capacity. So, we have not used mass in the calculation 

Concept:

• 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 generally denoted by C.

\(Q = mCΔ T\)              

Where Q = heat required, ΔT =  change in temperature and m = mass of the body.

So Specific heat capacity (C) is given by

\({\rm{\;\;C\;}} = {\rm{\;}}\frac{Q}{{mΔ T{\rm{\;}}\;}}\)

EXPLANATION:

• As it is heat required for unit mass of the body so specific heat doesn’t depend on the mass of that body.
• Specific heat capacity of anybody/material is a property of that body/material. It is same for each and every molecule of that body.
• So it doesn’t depend on the shape of the body. Thus independent of mass and shape of the body.
• As per the definition of the specific heat, it is the heat required to raise the temperature by 1°C. Hence the specific heat doesn’t depend on the temperature of the body.

• Since material to material the required heat energy to raise the temperature by 1°C per unit mass will differ. Hence specific heat capacity depends on material of the body/substance.

• Specific heat doesn’t depend on heat given because required heat energy for unit mass per rise of 1℃ can be calculated from the given heat energy to get specific heat.

CONCEPT:

• Specific heat or specific heat capacity of a body: The amount of heat energy required to raise the temperature of a unit mass of a body through 1° C (or K) is called the specific heat of the material of the body.
  • It is generally denoted by C.

Q = mC∆T              
Where Q = heat required, ∆T = change in temperature and m = mass of the body.

• So specific heat capacity is,

\({\rm{C\;}} = {\rm{\;}}\frac{Q}{{mΔ T{\rm{\;}}\;}}\)

EXPLANATION:

Given - The mass of substance = m, temperature change = ΔT amount of heat absorbed or rejected by a substance = ΔQ 

• The amount of heat absorbed or rejected by a substance of mass 'm' and specific heat capacity 's', when it undergoes a temperature change 'ΔT' is equal to

⇒ Q = ms∆T     

• So specific heat capacity is,

\(\Rightarrow {\rm{s\;}} = {\rm{\;}}\frac{Q}{{mΔ T{\rm{\;}}\;}}\)

Concept:

The specific heat capacity (S) 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.

\(i.e.,~~S=\frac{\text{ }\!\!\Delta\!\!\text{ }Q}{m\text{ }\!\!\Delta\!\!\text{ }T}\)

And the unit of Specific heat is J/g °C or J/kg °C  and Cal/g °C

Calculation:

Given that,

Mass of wood, m = 1 kg

Heat absorbed by wood for increasing its temperature by  25°C to 150°C, ΔQ = 200 kJ = 200000 J 

Change in temperature, ΔT = 150 - 25 = 125 °C 

Hence, from the above explanation, we can see that specific heat of any material can be calculated as 

\(~~S=\frac{\text{ }\!\!\Delta\!\!\text{ }Q}{m\text{ }\!\!\Delta\!\!\text{ }T}\)

\(~~S=\frac{\text{ }\Delta \text{ }Q}{m\text{ }\Delta \text{ }T}=\frac{\text{ }200000}{\text{1}\times 125}=1600J/kg{}^\circ C\)

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 C_p and C_v 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 C_p and C_v with a degree of freedom is given by

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

Triatomic gas has  6 degrees of freedom : 3 translational and 3 rotational

\(\gamma = 1 + \frac{2}{6} = 1 + \frac{1}{3} = \frac{4}{3} = 1.33\)

CONCEPT:

• 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 joule/g-K in the SI unit.
  • 1 calorie = 4.186 Joule.

s = (Q /m Δ t)

Where [s= specific heat of a substance, Q= Heat supplied to the object, m= mass of the substance, Δ t = change in temperature].

CALCULATION:

• 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.
• Hence option 4 is correct.

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.

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

• Internal energy (U) of a system is the total energy possessed by the system due to molecular motion and molecular configuration.
• Equipartition of energy states that the average energy of molecule in a gas associated with each degree of freedom is \(\frac{1}{2}k{T^2}\) where k is the Boltzmann constant and T is its absolute temperature.
• According to the equipartition theorem, the average energy of a molecule in a monoatomic gas is \(\frac{3}{2}kT\) as the degree of freedom is 3.
• According to the equipartition theorem, the average energy of a molecule in a diatomic gas is \(\frac{5}{2}kT\) if the molecules translate and rotate but do no vibrate, and is \(\frac{7}{2}kT\) if they vibrate also.

Explanation-

Now consider a sample of amount n moles of ideal gas. The total number of molecules is nN_A where N_A  is the Avogadro number. If the gas is diatomic, the internal energy of the gas is-

\(U = n{N_A}\left( {\frac{5}{2}kT} \right) = n\frac{5}{2}RT\)

If molecules do not vibrate. In this case,

\({C_v} = \frac{1}{n}\frac{{dU}}{{dT}} = \frac{5}{2}R\;\)

And

\({C_p} = {C_v} + R = \frac{5}{2}RT + R = \frac{7}{2}R\)

Then,

\(\gamma = \frac{{{C_p}}}{{{C_v}}} = \frac{{\frac{7}{2}R}}{{\frac{5}{2}R}} = \frac{7}{5} = 1.40\)

CONCEPT:

• Specific heat or specific heat capacity (C): The amount of heat required for a unit mass of body to raise the temperature by 1 degree Celsius is called specific heat capacity of that body.

\(Q = mCΔ T\)              

Where Q = heat required, ΔT =  change in temperature and m = mass of the body.

CALCULATION:

Given that:

Mass (m) = 10 g = 10/1000 kg = 0.01 kg

Initial temperature (T1) = 0 °C

Final temperature (T2) = 100 °C

Heat absorbed (Q) =?

C = 140 J/(kg K)

Use the formula:

\(Q = mCΔ T\)

Q = 0.01 × 140 × (100 - 0) = 140 J 

So option 4 is correct.

CONCEPT:

• Specific heat or specific heat capacity (C): The amount of heat required for a unit mass of body to raise the temperature by 1 degree Celsius is called specific heat capacity of that body.

\(Q = mCΔ T\)              

Where Q = heat required, ΔT =  change in temperature and m = mass of the body.

CALCULATION:

Given that:

Mass (m) = 100 g = 100/1000 kg = 0.1 kg

Initial temperature (T₁) = 20 °C

Heat given (Q) = 2250 J

C = 450 J/(kg K)

Use the formula:

\(Q = mCΔ T\)

2250 = 0.1 × 450 × ΔT

So ΔT = 2250/(0.1 × 450) = 2250/45 = 50 °C

Since ΔT = Final temperature (T₂) - Initial temperature (T₁) = T₂ - 20 °C = 50 °C

⇒ T₂ = 70 °C

Hence option 3 is correct.