2 kg of ice at -20$^{\circ}$C is mixed with 5 kg of water at 20$^{\circ}$C in an insulating vessel having a negligible heat capacity. Calculate the final mass of water remaining in the container. It is given that the specific heats of water and ice are 1 kcal/kg / $^{\circ}$C and 0.5 kcal/kg /$^{\circ}$C while the latent heat of fusion of ice is 80 kcal/kg
- 7 kg
- 6 kg
- 4 kg
- 2 kg
The Correct Option is B
Approach Solution - 1
Heat released by 5 kg of water when its temperature falls from 20\(^{\circ}\)C to 0\(^{\circ}\)C is
\(Q_1 =mc \Delta \theta =(5)(10^3)(20-0)=10^5 cal\)
when 2 kg ice at -20\(^{\circ}\)C comes to a temperature of 0\(^{\circ}\)C, it takes an energy
\(Q_2 =mc\Delta \theta =(2)(500)(20) =0.2 \times 10^{5}cal\)
The remaining heat \(Q=Q_1-Q_2 =0.8 \times 1^5\) cal will melt a mass m of the ice, where
\(\, \, \, \, \, \, \, \, \, \, \, m=\frac{Q}{L} =\frac{0.8 \times 10^5}{80 \times10^3}=1kg\)
So, the temperature of the mixture will be 0\(^{\circ}\)C, mass of water in it is 5 + 1 = 6 kg and mass of ice is 2 - 1 = 1 kg.
Approach Solution -2
At first, the ice absorbs heat to elevate its temperature to0∘C, after which it starts melting. Let's denote mi as the initial mass of the ice, mi′ as the mass of ice that melts, and mW as the initial mass of water. According to the Law of Mixture, the heat gained by the ice equals the heat lost by the water, leading to the equation:
\(m_i \times c \times (20) + m_i' \times L = m_W \times c_W \times (20)\)
This simplifies to:
\(2 \times 0.5 \times (20) + m_i' \times 80 = 5 \times 1 \times 20\)
Which further simplifies to:
\(𝑚𝑖′=1 kg\)
Therefore, the final mass of water is the initial mass of water plus the mass of ice that melts, which equals 5+1=6 kg.
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Concepts Used:
Thermodynamics
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
Important Terms
System
A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.
There are three types of systems:
- Isolated System – An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System – In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.
Thermodynamic Process
A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.
There are four types of thermodynamic process that have their unique properties, and they are:
- Adiabatic Process – A process in which no heat transfer takes place.
- Isochoric Process – A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process – A process in which no change in pressure occurs.
- Isothermal Process – A process in which no change in temperature occurs.
Laws of Thermodynamics
Zeroth Law of Thermodynamics
The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.
First Law of Thermodynamics
The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.
Second Law of Thermodynamics
The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.
Third Law of Thermodynamics
Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.