(specific heat of ice = $2100 \text{ J/Kg.}^\circ\text{C}$, specific heat of water = $4200 \text{ J/Kg.}^\circ\text{C}$, latent heat of fusion of ice = $3.36 \times 10^5 \text{ J/Kg}$)
Calorimetry Problem: Water Temperature Decrement Calculation
This problem involves calculating the change in water temperature when ice is added, using the principles of heat transfer and calorimetry.
Given Parameters for Heat Transfer
- Mass of ice, $m_i = 10 \text{ kg}$
- Initial temperature of ice, $T_{i,initial} = -10^\circ\text{C}$
- Specific heat of ice, $c_i = 2100 \text{ J/Kg.}^\circ\text{C}$
- Mass of water, $m_w = 100 \text{ kg}$
- Initial temperature of water, $T_{w,initial} = 25^\circ\text{C}$
- Specific heat of water, $c_w = 4200 \text{ J/Kg.}^\circ\text{C}$
- Latent heat of fusion of ice, $L_f = 3.36 \times 10^5 \text{ J/Kg}$
Heat Gained by Ice Stages
The ice must undergo several stages to reach thermal equilibrium. Let $T_f$ be the final equilibrium temperature.
- Heating ice to $0^\circ\text{C}$: The heat required is $Q_1$. $Q_1 = m_i c_i (0^\circ\text{C} - T_{i,initial}) = 10 \text{ kg} \times 2100 \text{ J/Kg.}^\circ\text{C} \times (0^\circ\text{C} - (-10^\circ\text{C})) = 10 \times 2100 \times 10 = 210,000 \text{ J}$
- Melting ice at $0^\circ\text{C}$: The heat required is $Q_2$. $Q_2 = m_i L_f = 10 \text{ kg} \times 3.36 \times 10^5 \text{ J/Kg} = 3,360,000 \text{ J}$
- Warming melted ice (now water) to $T_f$: The heat required is $Q_3$. $Q_3 = m_i c_w (T_f - 0^\circ\text{C}) = 10 \text{ kg} \times 4200 \text{ J/Kg.}^\circ\text{C} \times T_f = 42000 T_f \text{ J}$
Total heat gained by the ice is $Q_{gain} = Q_1 + Q_2 + Q_3$.
$Q_{gain} = 210,000 \text{ J} + 3,360,000 \text{ J} + 42000 T_f = 3,570,000 + 42000 T_f \text{ J}$Heat Lost by Water Calculation
The water cools from its initial temperature $T_{w,initial}$ to the final temperature $T_f$. The heat lost is $Q_{lost}$.
$Q_{lost} = m_w c_w (T_{w,initial} - T_f) = 100 \text{ kg} \times 4200 \text{ J/Kg.}^\circ\text{C} \times (25^\circ\text{C} - T_f)$ $Q_{lost} = 420000 (25 - T_f) = 10,500,000 - 420000 T_f \text{ J}$Calorimetry Equation Application
According to the principle of calorimetry, the heat lost by the water must equal the heat gained by the ice (assuming no heat exchange with surroundings).
$Q_{lost} = Q_{gain}$ $10,500,000 - 420000 T_f = 3,570,000 + 42000 T_f$Final Temperature Determination
Solve the equation for $T_f$:
$10,500,000 - 3,570,000 = 420000 T_f + 42000 T_f$ $6,930,000 = 462000 T_f$ $T_f = \frac{6,930,000}{462000} = 15^\circ\text{C}$Water Temperature Decrement Result
The decrement in the temperature of the water is the difference between its initial and final temperatures.
$\Delta T_w = T_{w,initial} - T_f = 25^\circ\text{C} - 15^\circ\text{C} = 10^\circ\text{C}$Similar Questions
- Which of the following best represents the temperature versus heat supplied graph for water, in the range of $-20^\circ\text{C}$ to $120^\circ\text{C}$ ?
- The volume of an ideal gas increases 8 times and temperature becomes $(1/4)^{\text{th}}$ of initial temperature during a reversible change. If there is no exchange of heat in this process ($\Delta Q = 0$) then identify the gas from the following options (Assuming the gases given in the options are ideal gases):
Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points $A$ and $F$ are maintained at $100^\circ\text{C}$ and $40^\circ\text{C}$ respectively. Given the thermal conductivity of rod x is three times of that of rod y, the temperature at junction points $B$ and $E$ are (close to):
- Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is $32 \times 10^{18}$ /s then collision frequency in gas A is _________ /s.
- An insulated cylinder of volume $60 \text{ cm}^3$ is filled with a gas at $27^\circ\text{C}$ and 2 atmospheric pressure. Then the gas is compressed making the final volume as $20 \text{ cm}^3$ while allowing the temperature to rise to $77^\circ\text{C}$. The final pressure is _________ atmospheric pressure.
- A brass wire of length 2 m and radius 1 mm at $27^\circ\text{C}$ is held taut between two rigid supports. Initially it was cooled to a temperature of $-43^\circ\text{C}$ creating a tension $T$ in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to $1.4T$, is ______ $^\circ\text{C}$.
- A gas of certain mass filled in a closed cylinder at a pressure of 3.23 kPa has temperature $50^\circ\text{C}$. The gas is now heated to double its temperature. The modified pressure is ______ Pa.
10 mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from $P_1$ to $P_2$ is $\alpha \text{ Joule}$ ($P_1 = 21.7 \text{ Pa}$ and $P_2 = 30 \text{ Pa}, C_v = 21 \text{ J/K.mol}, R = 8.3 \text{ J/mol.K}$). The value of $\alpha$ is _______.
- When $300 \text{ J}$ of heat given to an ideal gas with $C_p = \frac{7}{2} R$ its temperature raises from $20^\circ\text{C}$ to $50^\circ\text{C}$ keeping its volume constant. The mass of the gas is (approximately) _______ g. ($R = 8.314 \text{ J/mol.K}$)
- An aluminium and steel rods having same lengths and cross-sections are joined to make total length of $120 \text{ cm}$ at $30^\circ\text{C}$. The coefficient of linear expansion of aluminium and steel are $24 \times 10^{-6} /^\circ\text{C}$ and $1.2 \times 10^{-5} /^\circ\text{C}$, respectively. The length of this composite rod when its temperature is raised to $100^\circ\text{C}$, is ____________ $\text{cm}$.
Important Questions from Heat and Thermodynamics
- Which of the following best represents the temperature versus heat supplied graph for water, in the range of $-20^\circ\text{C}$ to $120^\circ\text{C}$ ?
- The volume of an ideal gas increases 8 times and temperature becomes $(1/4)^{\text{th}}$ of initial temperature during a reversible change. If there is no exchange of heat in this process ($\Delta Q = 0$) then identify the gas from the following options (Assuming the gases given in the options are ideal gases):
Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points $A$ and $F$ are maintained at $100^\circ\text{C}$ and $40^\circ\text{C}$ respectively. Given the thermal conductivity of rod x is three times of that of rod y, the temperature at junction points $B$ and $E$ are (close to):
- Consider two boxes containing ideal gases A and B such that their temperatures, pressures and number densities are same. The molecular size of A is half of that of B and mass of molecule A is four times that of B. If the collision frequency in gas B is $32 \times 10^{18}$ /s then collision frequency in gas A is _________ /s.
- An insulated cylinder of volume $60 \text{ cm}^3$ is filled with a gas at $27^\circ\text{C}$ and 2 atmospheric pressure. Then the gas is compressed making the final volume as $20 \text{ cm}^3$ while allowing the temperature to rise to $77^\circ\text{C}$. The final pressure is _________ atmospheric pressure.
