Thermal expansion MCQ Quiz - Objective Question with Answer for Thermal expansion - Download Free PDF

The Correct answer is 6500 calorie.

Key Points

• To calculate the total heat removed from water, we need to consider two stages:
  • Cooling the water from 50°C to 0°C.
  • Freezing the water at 0°C into ice.
• The formula to calculate heat required to cool water is: Q = m × c × ΔT, where:
  • m is the mass of the water (given as 50 g).
  • c is the specific heat of water (1 cal/g/°C).
  • ΔT is the temperature change (50°C - 0°C = 50°C).
• Substituting the values: Q = 50 × 1 × 50 = 2500 calories.
• For freezing water into ice, the formula is: Q = m × L, where:
  • L is the latent heat of fusion of ice (80 cal/g).
• Substituting the values: Q = 50 × 80 = 4000 calories.
• Total heat removed = 2500 + 4000 = 6500 calories.

Additional Information

• Latent Heat of Fusion:
  • The latent heat of fusion is the amount of heat energy required to change the state of a substance from solid to liquid (or vice versa) without changing its temperature.
  • For water, it is 80 cal/g, which means 80 calories of heat energy are required to convert 1 gram of ice into water at 0°C.
• Specific Heat:
  • The specific heat is the amount of heat energy required to raise the temperature of 1 gram of a substance by 1°C.
  • For water, the specific heat is 1 cal/g/°C, which is relatively high compared to most substances.
• Importance of Cooling and Freezing:
  • Cooling water to 0°C involves removing heat based on the temperature difference and specific heat.
  • Freezing water to ice requires additional heat removal, calculated using the latent heat of fusion.

Calculation:

Let the thermal conductivity is k.
dQ/dt = -kA dT/dx = -k[2π(b-x) × dT/dx]

On integrating with respect to dx, we get:

∫₀^b-a dQ/dt × 1/2π(b-x) dx = ∫_a^b-a -kl dT/dx × dx

⇒ dQ/dt × (-1/2π)×  ln(b-x) ^b-a₀ = -kl ΔT

dQ/dt ln(b/a) = -2π k l (T-T₀)

Since s= C/V and dQ= C dT

⇒ (πa²)s dT/dt [ln(b/a)] = -2π k (T - T₀)

⇒ On integrating with respect to t we get:

πa² s ln [b/a]  ∫ dT/(T-T₀) = -2π k t

⇒ πa² s ln [b/a]  ln[(T0-T1)/(T0-T2)]  = 2π k t

for t = ln2 / 2 and b=2a

k= a2 s ln[(T0-T1)/(T0-T2)]

ΔL = L α ΔT

The tension in the ring is T.

T / A = (ΔL / L) × Y

T = α ΔT Y S

So, F = 2T

F = 2 α ΔT Y S

putting the value

⇒ F=2 

Concept:

Thermal Expansion Analysis:

When a material is heated, it expands. The expansion can be linear, superficial, or cubical.

The coefficient of cubical expansion (γ) is related to the coefficient of superficial expansion (β).

\( \gamma = 3\beta \)

Where:

\( \beta \) is the coefficient of superficial expansion.

\( \gamma \) is the coefficient of cubical expansion.

Calculation:

Given that the coefficient of cubical expansion (γ) is 'x' times the coefficient of superficial expansion (β), we have:

\( \gamma = x \beta \)

From the relationship between γ and β, we know:

\( \gamma = 3\beta \)

Equating the two equations:

\( x \beta = 3 \beta \)

Dividing both sides by β, we get:

\( x = 3 \)

∴ The value of 'x' is 3.

\(l_b = l_0 (1 + \alpha_b \Delta T)\)

\(l_c = l_0 (1 + \alpha_c \Delta T)\)

\(l_c = R \theta\)

\(l_b + (R + d)\theta\)

Thus \(\displaystyle \frac{R + d}{R} = \frac{1 + \alpha_b \Delta T}{1 + \alpha_c \Delta T}\)

or \(1 + \displaystyle\frac{d}{R} = 1 + (\alpha_b - \alpha_c) \Delta T\)

(Thus correct choices are (b) and (d))

Explanation:

Coefficient of volume expansion:

• It is the measure of the fractional change of size per unit change in the temperature with the surrounding pressure being constant.
• It affects the volume of the particular body.
• This coefficient is most applicable to fluids.
• It generally depends on the temperature as the coefficient increases with the subsequent increase in temperature.
• For a small temperature change, ∆T, the fractional change in volume, ∆V/V, is directly proportional to ∆T.​
  • ∆V/V = αv ∆T, where αv is the coefficient of volume expansion.

Explanation:

Coefficient of volume expansion:

• It is the measure of the fractional change of size per unit change in the temperature with the surrounding pressure being constant.
• It affects the volume of the particular body.
• This coefficient is most applicable to fluids.
• It generally depends on the temperature as the coefficient increases with the subsequent increase in temperature.
• For a small temperature change, ∆T, the fractional change in volume, ∆V/V, is directly proportional to ∆T.​
  • ∆V/V = αv ∆T, where αv is the coefficient of volume expansion.

CONCEPT:

• Properties of matter: At different temperature, matter behaves differently, but most common properties are-
  • When the temperature decreases matter contracts (molecules lose energy).
  • When temperature increases matter expands (molecules gain energy).
• Water: In the case of water there is an analogy with temperature and volume.
  • From 4° C to 100° C water expands when heated.
  • At 4° C water became denser and having minimum volume.
  • From 4° C to 0° C it again expands.

EXPLANATION:

• In winter: When the temperature goes below 4° the metal pipes get contracts and the water got expands (becoming ice).
• In this way, metal pipes get busted as water expanded when freezes.

option (1)

CONCEPT:

• Thermal Expansion: Whenever the solids, liquids, and gases are heated at some temperature, they start changing their shape and that process is known as Thermal Expansion.
  • Thermal Expansion made up of two words Thermal - Heat and Expansion - Change in the shape of the body.

EXPLANATION:

• All solids, liquids, and gases consist of atoms and molecules.
• On giving any heat to these atoms and molecules start changing the shape and size due to high kinetic energy between the particle.
• This results in the thermal expansion of the solid.

Additional Information

• Apparent Expansion: When a liquid is heated, its container also expanded and this observed expansion is known as Apparent Expansion.
• Real Expansion: It is the combination of Apparent expansion of the liquids and volume expansion of the container.
• The real expansion of the liquid = Apparent expansion of the liquid + Volume expansion of the container.

CONCEPT:

• Dimensions of an object changes as it is subjected to heat, the change in dimension are of 3 types:
  1. Linear expansion
  2. Area expansion
  3. Volume expansion
• Linear expansion is when the solid rod of initial length 'L' is heated through a temperature ΔT, the newly increased length of  the rod L' is given by

⇒ L'  = L(1 + α Δ T)

Where L' = The newly increased length, α = Coefficient of linear expansion, and ΔT = Difference in temperature.

• The coefficient of linear expansion (α) is given by

\(\alpha = \frac{L' - L}{LΔ T}\)

CALCULATION :

• From above it is clear that, the coefficient of linear expansion (α) is given by

\(\Rightarrow \alpha = \frac{L' - L}{LΔ T}\)

⇒ ΔL = L' - L

\(\Rightarrow \alpha = \frac{\Delta L}{LΔ T}\)

• Option 4 is the answer

NOTE:

• Area expansion is also known as superficial expansion. 
• When a solid of known surface area S is heated through a temperature difference Δ T, then the final surface area is given by

⇒ S' = S(1+βΔ T)

S' = The new surface area, S = Old surface area, β = Coefficient of area expansion, and Δ T = Temperature difference

• The coefficient of the  area  expansion is given by 

\(\beta = \frac{S^{'}-S}{S\times Δ T}\)

• Volume expansion of a solid is when a solid of known volume V is heated through a temperature difference ΔT, The final volume is given by

V' = V(1+γΔT)

• The coefficient of volume expansion is given by

\(\gamma=\frac{V^{'}- V}{V\times Δ T}\)

CONCEPT:

• Thermal expansion: When the temperature of any object is changed then the tendency of changing the shape, area, and volume of the body is called the thermal expansion of that object.

There are three types of thermal expansion:

The relation between α, β, and γ is given by:

β = 2α, and γ = 3α

• Density (d): The mass per unit volume is called density.

Density (d) = Mass (m)/Volume (V)

EXPLANATION:

• Since the radius increment is a linear expansion, which is directly proportional to the temperature change. 
• Similarly, the area change and volume change are directly proportional to the temperature change.

Density change (Δ d) = Mass (m)/Change in volume (ΔV)

• Since the density change is inversely proportional to volume change, so the density will have a minimum percentage increase. Hence option 4 is correct.

CONCEPT:

• Dimensions of an object changes as it is subjected to heat, the change in dimension are of 3 types:
  1. Linear expansion
  2. Area expansion
  3. Volume expansion
• Linear expansion is when the solid rod of initial length 'L' is heated through a temperature ΔT, the newly increased length of  the rod L' is given by

⇒ L'  = L(1 + α Δ T)

Where L' = The newly increased length, α = Coefficient of linear expansion, and ΔT = Difference in temperature.

• The coefficient of linear expansion (α) is given by

\(α = \frac{L' - L}{LΔ T} =\frac{Δ L}{LΔ T}\)

• The percentage of linear expansion is given

\(\frac{Δ L}{L}×100\)

CALCULATION:

Given -  α = 1.8 x 10-5 K-1 , ΔL = 0.45 mm =0.45× 10^-3 M, L = 1 m

• The coefficient of linear expansion (α) is given by

\(\Rightarrow α = \frac{L' - L}{LΔ T} =\frac{Δ L}{LΔ T}\)

Above equation can be rewritten for ΔT as

\(\Rightarrow\Delta T=\frac{Δ L}{L\alpha}\)

Substituting the given values in equation 2

\(\Rightarrow\Delta T=\frac{Δ L}{L\alpha} = \frac{0.45 \times 10^{-3}}{1\times1.8\times10^5} = 25^0C\)

• The temperature of the brass scale is expanded by 25°C.

NOTE:

• Area expansion is also known as superficial expansion. 
• When a solid of known surface area S is heated through a temperature difference Δ T, then the final surface area is given by

⇒ S' = S(1+ βΔ T)

S' = The new surface area, S = Old surface area, β = Coefficient of area expansion, and Δ T = Temperature difference

• Volume expansion of a solid is when a solid of known volume V is heated through a temperature difference ΔT, The final volume is given by

V' = V(1+ γΔT)

Concept:

Expansion of solids when heated is known as thermal expansion of Solid.

Thermal expansion of solids can be further classified into three types based on their change in dimension.

1. Linear Expansion: When the expansion of solid is linear when heated, such expansion is known as linear expansion

and coefficient of linear expansion,

\(α =\frac{Increase~in~length}{Original~length\times change~in~temperature~}=\frac{\text{ }\!\!\Delta\!\!\text{ }L}{L\times \text{ }\!\!\Delta\!\!\text{ }T}\)

2. Areal/Superficial Expansion: When the expansion of solid expands along two dimensions, i.e., in case of expansion of lamina both length and breadth will expand when heated such expansion is known as Superficial expansion

Coefficient of Superficial expansion,

\(β =\frac{\text{ }\!\!\Delta\!\!\text{ }A}{A\times \text{ }\!\!\Delta\!\!\text{ }T}\)

3.Volumetric Expansion or Cubical Expansion:

When the expansion of solids expands along three dimensions, i.e., in the case of expansion of lamina, both length, height, and breadth will expand when heated, such expansion is known as Volumetric expansion or Cubical Expansion.

Coefficient of Volumetric expansion,

\(γ =\frac{\text{ }\!\!\Delta\!\!\text{ }V}{V\times \text{ }\!\!\Delta\!\!\text{ }T}\)

Relationship between α, β, and γ

Using error theorem, we can say that, β = 2α, γ = 3α, and

α : β : γ = 1 : 2 : 3

Explanation:

Since the coefficient of linear expansion is in one direction and areal in two dimensions followed by three-dimension in case of volume expansion.

So, to compensate that factor the relation between them will be like.

α = β /2 = γ /3

Multiplying with 6 we get,

6α = 3β = 2γ 

CONCEPT:

• Thermal Conductivity: When one end of a metal rod is heated, heat flows by conduction from the hot end to the cold end.
• In this process, each cross-section of the rod receives some heat from the adjacent cross-section towards the hot end.
• It is found that the amount of heat Q that flows from hot to cold face during steady-state is 

\(Q = \frac{{KA\left( {{T_1} - {T_2}} \right)t}}{x}\)

Where K = Coefficient of thermal conductivity of the material.

• Rate of conduction of heat energy is given by:

\(\frac{{dQ}}{t} = \frac{{KA\left( {{T_1} - {T_2}} \right)}}{x} = KA\frac{{{\bf{\Delta }}T}}{x}\)

EXPLANATION:

• The rate of flow of heat through a metallic bar is given by:

\(\Rightarrow \frac{{dQ}}{t} = \frac{{KA\left( {{T_1} - {T_2}} \right)}}{L} = KA\frac{{{\bf{\Delta }}T}}{L}\)

Where K = thermal conductivity, L = length of the bar, ΔT = temperature difference at the ends of the bar, and A = area of cross-section of the bar

Therefore option 3 is correct.

Concept:

• Thermal Expansion: Whenever the solids, liquids, and gases are heated at some temperature, they start changing their shape and that process is known as thermal expansion.
  • Thermal expansions are of mainly three types.

Explanation:

​Thermal expansion of solids can be further classified into three types based on their change in dimension.

1. Linear Expansion: When the expansion of solid is linear when heated, such expansion is known as linear expansion

and coefficient of linear expansion,

\(\alpha =\frac{Increase~in~length}{Original~length\times change~in~temperature~}=\frac{\text{ }\!\!\Delta\!\!\text{ }L}{L\times \text{ }\!\!\Delta\!\!\text{ }T}\)

2. Areal/Superficial Expansion: When the expansion of solid expands along two dimensions, i.e., in case of expansion of lamina both length and breadth will expand when heated such expansion is known as Superficial expansion

Coefficient of Superficial expansion,

\(\beta =\frac{\text{ }\!\!\Delta\!\!\text{ }A}{A\times \text{ }\!\!\Delta\!\!\text{ }T}\)

3.Volumetric Expansion or Cubical Expansion:

When the expansion of solids expands along three dimensions, i.e., in the case of expansion of lamina, both length, height, and breadth will expand when heated, such expansion is known as Volumetric expansion or Cubical Expansion.

Coefficient of Volumetric expansion,

\(\gamma =\frac{\text{ }\!\!\Delta\!\!\text{ }V}{V\times \text{ }\!\!\Delta\!\!\text{ }T}\)

Relationship between α, β, and γ

Using error theorem, we can say that, β=2α, γ=3α, and α:β:γ=1:2:3