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Solved Problems on Specific Heat, Latent Heat and Entropy:- Problem 1:- A spherical constant temperature heat source of radius r_{1} is at the center of a uniform solid sphere of radius r_{2}. Find out the rate which is proportional to heat transferred through the surface of the sphere. Solution:- The rate H at which heat is transferred through the slab is, (a) directly proportional to the area (A) available. (b) inversely proportional to the thickness of the slab Δx. (c) directly proportional to the temperature difference ΔT. So, H = kA ΔT/ Δx Where k is the proportionality constant and is called thermal conductivity of the material. From above we know that, the rate H at which heat is transferred through the slab is directly proportional to the area (A) available. Area A of solid sphere is defined as, A = 4πr^{2} Here r is the radius of sphere. So, the area A_{1} of uniform small solid sphere having radius r_{1} will be, A_{1} = 4πr_{1}^{2} And, the area A_{2} of uniform large solid sphere having radius r_{2} will be, A_{2} = 4πr_{2}^{2} Thus the area A from which heat is transferred through the surface of the sphere will be the difference of area of uniform large solid sphere A_{2} and small solid sphere A_{1}. So, A = A_{2} - A_{1} = 4πr_{2}^{2} - 4πr_{1}^{2} = 4π (r_{2}^{2} - r_{1}^{2}) Since the rate H at which heat is transferred through the slab is directly proportional to the area (A) available, therefore the rate at which heat is transferred through the surface of the sphere is proportional to r_{2}^{2} - r_{1}^{2}. Problem 2:- What mass of steam at 100°C must be mixed with 150 g of ice at 0°C, in a thermally insulated container, to produce liquid water at 50°C. Concept:- The heat capacity per unit mass of a body, called specific heat capacity or usually just specific heat, is characteristic of the material of which the body is composed. c = C/m = Q/mΔT So, Q = c mΔT Here, the heat transferred is Q, specific heat capacity is c, mass is m and the temperature difference is ΔT. The amount of heat per unit mass that must be transferred to produce a phase change is called the heat of transformation or latent heat L for the process. The total heat Q transferred in a phase change is then, Q = Lm Here m is the mass of the sample that changes phase. Solution:- The heat given off the steam Q_{s} will be equal to, Q_{s} = m_{s}L_{v}+ m_{s}c_{w}ΔT Here, mass of steam is m_{s}, latent heat vaporization is L_{v}, specific heat capacity of water is c_{w} and the temperature difference is ΔT. The heat taken in by the ice Q_{i} will be equal to, Q_{i} = m_{i}L_{f}+ m_{i}c_{w}ΔT Here, mass of ice is m_{i}, latent heat fusion is L_{f}, specific heat capacity of water is c_{w} and the temperature difference is ΔT. Heat given off the steam Q_{s} is equal to the heat taken in by the ice Q_{i}. So, Q_{s} = Q_{i} m_{s}L_{v}+ m_{s}c_{w}ΔT = m_{i}L_{f}+ m_{i}c_{w}ΔT m_{s}(L_{v}+ c_{w}ΔT) = m_{i}(L_{f}+ c_{w}ΔT) m_{s} = m_{i}(L_{f}+ c_{w}ΔT)/ (L_{v}+ c_{w}ΔT) To obtain the mass of the steam at 100^{ °}C must be mixed with 150 g of ice at 0 ^{°}C, substitute 150 g for mass of ice m_{i}, 333×10^{3} J/kg for L_{f}, 4190 J/kg.K for c_{w}, 50^{°} C for ΔT, 2256×10^{3} J/kg for L_{v} in the equation m_{s} = m_{i}(L_{f}+ c_{w}ΔT)/ (L_{v}+ c_{w}ΔT), we get, m_{s} = m_{i}(L_{f}+ c_{w}ΔT)/ (L_{v}+ c_{w}ΔT) =(150 g)[(333×10^{3} J/kg) +(4190 J/kg.K) (50^{°} C)]/ [(2256×10^{3} J/kg)+ (4190 J/kg.K) (50^{°} C)] =(150 g×(10^{-3} kg/1g))[(333×10^{3} J/kg) +(4190 J/kg.K) (50+273)K]/ [(2256×10^{3} J/kg)+ (4190 J/kg.K) (50+273)K] = 0.033 kg From the above observation we conclude that, the mass of steam at 100^{ °}C must be mixed with 150 g of ice at 0 ^{°}C would be 0.033 kg. Problem 3:- (a) Compute the possible increase in temperature for water going over Niagara Falls, 49.4 m high. (b) What factors would tend to prevent this possible rise? Concept:- Work done W is defined as, W = mgΔy Here m is the mass, g is the free fall acceleration and Δy is the increase in height. The specific heat capacity c of a material is equal to the heat capacity C per unit mass m of the body. So, c = C/m = Q/mΔT (Since, C = Q/ΔT ) Here ΔT is the increase in temperature. So, Q = mcΔT As, |Q| = |W|, mcΔT = mgΔy So, ΔT = gΔy/c Solution:- (a) To obtain the possible increase in temperature ΔT, substitute 9.81 m/s^{2} for g, 49.4 m for Δy and 4190 J/kg.K for specific heat capacity of water c in the equation ΔT = gΔy/c, ΔT = gΔy/c = (9.81 m/s^{2}) (49.4 m)/ (4190 J/kg.K) = (9.81 m/s^{2}) (49.4 m)/ (4190 J/kg.K) (1 kg.m^{2}/s^{2} /1 J) = 0.116 K From the above observation we conclude that, the possible increase in temperature ΔT would be 0.116 K (b) The above expression is valid only there is no loss energy. But due to the viscosity of the water, some of the energy is lost in rising against the gravity. So, the factor which prevents the water from rising is the viscosity. Problem 4:- In a certain solar house, energy from the Sun is stored in barrels filled with water. In a particular winter stretch of five cloudy days, 5.22 GJ are needed to maintain the inside of the house at 22.0°C. Assuming that the water in the barrels is at 50.0°C, what volume of water is required? Concept:- Heat Q that must be given to a body of mass m, whose material has a specific heat c, to increase its temperature from initial temperature T_{i} to final temperature T_{f} is, Q = mc (T_{f} - T_{i}) So, m = Q/ c (T_{f} - T_{i}) Density ρ is equal to mass m per unit volume V. So, ρ = m/V So volume V will be, V = m/ρ Solution:- To find the volume water, first we have to find out the mass of water which is required to transfer 5.22 GJ amount of heat energy. To find the mass m of water, substitute 5.22 GJ for Q, 4190 J/kg. K for specific heat capacity c of water, 50.0 ^{°} C for T_{f} and 22.0 ° C for T_{i} in the equation m = Q/ c (T_{f} - T_{i}), m = Q/ c (T_{f} - T_{i}) = 5.22 GJ/(4190 J/kg. K) (50.0 ^{°} C-22.0 ° C) = (5.22 GJ) (10^{9} J/1 GJ)/(4190 J/kg. K) ((50.0+273) K –(22.0+273) K) = (5.22 ×10^{9} J)/(4190 J/kg. K) (28 K) = 4.45×10^{4} kg To obtain the volume V of water, substitute 4.45×10^{4} kg for mass m and 998 kg/m^{3} for density ρ of water in the equation V = m/ρ, V = m/ρ = (4.45×10^{4} kg) / (998 kg/m^{3}) = 44.5 m^{3} From the above observation we conclude that, the volume V of water will be 44.5 m^{3}. Problem 5:- A small electric immersion heater is used to boil 136 g of water for a cup of instant coffee. The heater is labeled 220 watts. Calculate the time required to bring this water from 23.5°C to the boiling point, ignoring any heat losses. Concept:- Heat Q that must be given to a body of mass m, whose material has a specific heat c, to increase its temperature from initial temperature T_{i} to final temperature T_{f} is, Q = mc (T_{f} - T_{i}) But time (t) is equal to the heat energy (Q) divided by power (P). t = Q/ P = mc (T_{f} - T_{i})/P Solution:- To obtain the time required to bring this water from 23.5^{°} C to the boiling point, substitute 136 g for mass of water m, 4190 J/kg. K for specific heat capacity of water c, 100^{°} C for final temperature T_{f} (boiling point of water), 23.5° C for initial temperature T_{i} and 220 watts for power P in the equation t = mc (T_{f} - T_{i})/P, t = mc (T_{f} - T_{i})/P = (136 g) (4190 J/kg. K) (100^{°} C - 23.5° C) / 220 W = (136 g×10^{-3} kg/1 g) (4190 J/kg. K) (100^{°} C - 23.5° C) / 220 W = (0.136 kg) (4190 J/kg. K) ((100+273) K – (23.5 + 273)K) / 220 W = (0.136 kg) (4190 J/kg. K) (373 K – 296.5 K) / 220 W = (0.136 kg) (4190 J/kg. K) (76.5 K) / 220 W = 198.15 s Rounding off to three significant figures, the time required to bring this water from 23.5^{°} C to the boiling point would be 198 s. Related Resources: You might like to refer some of the related resources listed below: Click here for the Detailed Syllabus of IIT JEE Physics. Look into the Sample Papers of Previous Years to get a hint of the kinds of questions asked in the exam. To read more, Buy study materials of Thermodynamics comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Chemistry here.
A spherical constant temperature heat source of radius r_{1} is at the center of a uniform solid sphere of radius r_{2}. Find out the rate which is proportional to heat transferred through the surface of the sphere.
The rate H at which heat is transferred through the slab is,
(a) directly proportional to the area (A) available.
(b) inversely proportional to the thickness of the slab Δx.
(c) directly proportional to the temperature difference ΔT.
So, H = kA ΔT/ Δx
Where k is the proportionality constant and is called thermal conductivity of the material.
From above we know that, the rate H at which heat is transferred through the slab is directly proportional to the area (A) available.
Area A of solid sphere is defined as,
A = 4πr^{2}
Here r is the radius of sphere.
So, the area A_{1} of uniform small solid sphere having radius r_{1} will be,
A_{1} = 4πr_{1}^{2}
And, the area A_{2} of uniform large solid sphere having radius r_{2} will be,
A_{2} = 4πr_{2}^{2}
Thus the area A from which heat is transferred through the surface of the sphere will be the difference of area of uniform large solid sphere A_{2} and small solid sphere A_{1}.
So, A = A_{2} - A_{1}
= 4πr_{2}^{2} - 4πr_{1}^{2}
= 4π (r_{2}^{2} - r_{1}^{2})
Since the rate H at which heat is transferred through the slab is directly proportional to the area (A) available, therefore the rate at which heat is transferred through the surface of the sphere is proportional to r_{2}^{2} - r_{1}^{2}.
What mass of steam at 100°C must be mixed with 150 g of ice at 0°C, in a thermally insulated container, to produce liquid water at 50°C.
The heat capacity per unit mass of a body, called specific heat capacity or usually just specific heat, is characteristic of the material of which the body is composed.
c = C/m
= Q/mΔT
So, Q = c mΔT
Here, the heat transferred is Q, specific heat capacity is c, mass is m and the temperature difference is ΔT.
The amount of heat per unit mass that must be transferred to produce a phase change is called the heat of transformation or latent heat L for the process. The total heat Q transferred in a phase change is then,
Q = Lm
Here m is the mass of the sample that changes phase.
The heat given off the steam Q_{s} will be equal to,
Q_{s} = m_{s}L_{v}+ m_{s}c_{w}ΔT
Here, mass of steam is m_{s}, latent heat vaporization is L_{v}, specific heat capacity of water is c_{w} and the temperature difference is ΔT.
The heat taken in by the ice Q_{i} will be equal to,
Q_{i} = m_{i}L_{f}+ m_{i}c_{w}ΔT
Here, mass of ice is m_{i}, latent heat fusion is L_{f}, specific heat capacity of water is c_{w} and the temperature difference is ΔT.
Heat given off the steam Q_{s} is equal to the heat taken in by the ice Q_{i}.
So, Q_{s} = Q_{i}
m_{s}L_{v}+ m_{s}c_{w}ΔT = m_{i}L_{f}+ m_{i}c_{w}ΔT
m_{s}(L_{v}+ c_{w}ΔT) = m_{i}(L_{f}+ c_{w}ΔT)
m_{s} = m_{i}(L_{f}+ c_{w}ΔT)/ (L_{v}+ c_{w}ΔT)
To obtain the mass of the steam at 100^{ °}C must be mixed with 150 g of ice at 0 ^{°}C, substitute 150 g for mass of ice m_{i}, 333×10^{3} J/kg for L_{f}, 4190 J/kg.K for c_{w}, 50^{°} C for ΔT, 2256×10^{3} J/kg for L_{v} in the equation m_{s} = m_{i}(L_{f}+ c_{w}ΔT)/ (L_{v}+ c_{w}ΔT), we get,
=(150 g)[(333×10^{3} J/kg) +(4190 J/kg.K) (50^{°} C)]/ [(2256×10^{3} J/kg)+ (4190 J/kg.K) (50^{°} C)]
=(150 g×(10^{-3} kg/1g))[(333×10^{3} J/kg) +(4190 J/kg.K) (50+273)K]/ [(2256×10^{3} J/kg)+ (4190 J/kg.K) (50+273)K]
= 0.033 kg
From the above observation we conclude that, the mass of steam at 100^{ °}C must be mixed with 150 g of ice at 0 ^{°}C would be 0.033 kg.
(a) Compute the possible increase in temperature for water going over Niagara Falls, 49.4 m high. (b) What factors would tend to prevent this possible rise?
Work done W is defined as,
W = mgΔy
Here m is the mass, g is the free fall acceleration and Δy is the increase in height.
The specific heat capacity c of a material is equal to the heat capacity C per unit mass m of the body.
So, c = C/m
= Q/mΔT (Since, C = Q/ΔT )
Here ΔT is the increase in temperature.
So, Q = mcΔT
As, |Q| = |W|,
mcΔT = mgΔy
So, ΔT = gΔy/c
(a) To obtain the possible increase in temperature ΔT, substitute 9.81 m/s^{2} for g, 49.4 m for Δy and 4190 J/kg.K for specific heat capacity of water c in the equation ΔT = gΔy/c,
ΔT = gΔy/c
= (9.81 m/s^{2}) (49.4 m)/ (4190 J/kg.K)
= (9.81 m/s^{2}) (49.4 m)/ (4190 J/kg.K) (1 kg.m^{2}/s^{2} /1 J)
= 0.116 K
From the above observation we conclude that, the possible increase in temperature ΔT would be 0.116 K
In a certain solar house, energy from the Sun is stored in barrels filled with water. In a particular winter stretch of five cloudy days, 5.22 GJ are needed to maintain the inside of the house at 22.0°C. Assuming that the water in the barrels is at 50.0°C, what volume of water is required?
Heat Q that must be given to a body of mass m, whose material has a specific heat c, to increase its temperature from initial temperature T_{i} to final temperature T_{f} is,
Q = mc (T_{f} - T_{i})
So, m = Q/ c (T_{f} - T_{i})
Density ρ is equal to mass m per unit volume V.
So, ρ = m/V
So volume V will be,
V = m/ρ
To find the volume water, first we have to find out the mass of water which is required to transfer 5.22 GJ amount of heat energy.
To find the mass m of water, substitute 5.22 GJ for Q, 4190 J/kg. K for specific heat capacity c of water, 50.0 ^{°} C for T_{f} and 22.0 ° C for T_{i} in the equation m = Q/ c (T_{f} - T_{i}),
m = Q/ c (T_{f} - T_{i})
= 5.22 GJ/(4190 J/kg. K) (50.0 ^{°} C-22.0 ° C)
= (5.22 GJ) (10^{9} J/1 GJ)/(4190 J/kg. K) ((50.0+273) K –(22.0+273) K)
= (5.22 ×10^{9} J)/(4190 J/kg. K) (28 K)
= 4.45×10^{4} kg
To obtain the volume V of water, substitute 4.45×10^{4} kg for mass m and 998 kg/m^{3} for density ρ of water in the equation V = m/ρ,
= (4.45×10^{4} kg) / (998 kg/m^{3})
= 44.5 m^{3}
From the above observation we conclude that, the volume V of water will be 44.5 m^{3}.
A small electric immersion heater is used to boil 136 g of water for a cup of instant coffee. The heater is labeled 220 watts. Calculate the time required to bring this water from 23.5°C to the boiling point, ignoring any heat losses.
But time (t) is equal to the heat energy (Q) divided by power (P).
t = Q/ P
= mc (T_{f} - T_{i})/P
To obtain the time required to bring this water from 23.5^{°} C to the boiling point, substitute 136 g for mass of water m, 4190 J/kg. K for specific heat capacity of water c, 100^{°} C for final temperature T_{f} (boiling point of water), 23.5° C for initial temperature T_{i} and 220 watts for power P in the equation t = mc (T_{f} - T_{i})/P,
t = mc (T_{f} - T_{i})/P
= (136 g) (4190 J/kg. K) (100^{°} C - 23.5° C) / 220 W
= (136 g×10^{-3} kg/1 g) (4190 J/kg. K) (100^{°} C - 23.5° C) / 220 W
= (0.136 kg) (4190 J/kg. K) ((100+273) K – (23.5 + 273)K) / 220 W
= (0.136 kg) (4190 J/kg. K) (373 K – 296.5 K) / 220 W
= (0.136 kg) (4190 J/kg. K) (76.5 K) / 220 W
= 198.15 s
Rounding off to three significant figures, the time required to bring this water from 23.5^{°} C to the boiling point would be 198 s.
Related Resources: You might like to refer some of the related resources listed below:
Click here for the Detailed Syllabus of IIT JEE Physics.
Look into the Sample Papers of Previous Years to get a hint of the kinds of questions asked in the exam.
To read more, Buy study materials of Thermodynamics comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Chemistry here.
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