Understanding Newton's Law of Cooling
This problem involves Newton's Law of Cooling, which describes how the rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings.
The integrated form of Newton's Law of Cooling is given by:
$ \ln\left(\frac{T_1 - T_s}{T_2 - T_s}\right) = kt $Where:
- $T_1$ is the initial temperature of the body.
- $T_2$ is the final temperature of the body.
- $T_s$ is the constant temperature of the surroundings.
- $k$ is the cooling constant.
- $t$ is the time taken for the temperature change.
Calculating the Cooling Constant (k)
We are given the first cooling interval:
- $T_1 = 60^\circ C$
- $T_2 = 50^\circ C$
- $T_s = 25^\circ C$
- $t = 10$ minutes
Plugging these values into the formula:
$ \ln\left(\frac{60 - 25}{50 - 25}\right) = k \times 10 $ $ \ln\left(\frac{35}{25}\right) = 10k $ $ \ln\left(\frac{7}{5}\right) = 10k $So, the cooling constant $k$ is:
$ k = \frac{1}{10} \ln\left(\frac{7}{5}\right) $Determining Temperature After Next 10 Minutes
Now, we need to find the temperature ($T_3$) after the next 10 minutes. The second interval starts from $T_2 = 50^\circ C$ and lasts for $t = 10$ minutes.
- Initial temperature for this interval: $T_2 = 50^\circ C$
- Final temperature for this interval: $T_3$ (unknown)
- Surrounding temperature: $T_s = 25^\circ C$
- Time interval: $t = 10$ minutes
Using the formula again:
$ \ln\left(\frac{T_2 - T_s}{T_3 - T_s}\right) = kt $Substitute the value of $k$ and the known temperatures:
$ \ln\left(\frac{50 - 25}{T_3 - 25}\right) = \left(\frac{1}{10} \ln\left(\frac{7}{5}\right)\right) \times 10 $ $ \ln\left(\frac{25}{T_3 - 25}\right) = \ln\left(\frac{7}{5}\right) $Equating the terms inside the logarithm:
$ \frac{25}{T_3 - 25} = \frac{7}{5} $Solve for $T_3$:
$ 25 \times 5 = 7 \times (T_3 - 25) $ $ 125 = 7T_3 - 175 $ $ 7T_3 = 125 + 175 $ $ 7T_3 = 300 $ $ T_3 = \frac{300}{7} $ $ T_3 \approx 42.857^\circ C $Final Temperature Approximation
The calculated temperature $T_3$ is approximately $42.857^\circ C$. Comparing this with the given options, the closest value is $43^\circ C$. Therefore, the temperature of the body after the next 10 minutes will be approximately $43^\circ C$.
Similar Questions
During the melting of a slab of ice at $273 \ K$ at atmospheric pressure:
- $\gamma_A$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. $\gamma_B$ is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If $\frac{\gamma_A}{\gamma_B} = \left(1 + \frac{1}{n}\right)$, then the value of n is _________.
- Match List - I with List - II.$\Delta Q$=Heat supplied
List - I List - II (A) Isobaric (I) $\Delta Q = \Delta W$ (B) Isochoric (II) $\Delta Q = \Delta U$ (C) Adiabatic (III) $\Delta Q = \text{zero}$ (D) Isothermal (IV) $\Delta Q = \Delta U + P\Delta V$
$\Delta W$ = Work done by the system
$\Delta U$=Change in internal energy
P = Pressure of the system
$\Delta V$ = Change in volume of the system
Choose the correct answer from the options given below : - There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K. If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K, at steady state the pressure in the vessels will be (in kPa).
Match the LIST-I with LIST-II Choose the correct answer from the options given below:
An ideal gas has undergone through the cyclic process as shown in the figure. Work done by the gas in the entire cycle is ________ $\times 10^{-1}$J.
(Take $\pi = 3.14$)
- An ideal gas exists in a state with pressure $P_0$, volume $V_0$. It is isothermally expanded to $4$ times of its initial volume($V_0$), then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is
Water falls from a height of $200 \text{ m}$ into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool.
(Take $g = 10 \text{ m/s}^2$, specific heat of water $= 4200 \text{ J/(kg K)}$)
A monoatomic gas having $\gamma = \frac{5}{3}$ is stored in a thermally insulated container and the gas is suddenly compressed to $\frac{1}{8}^{\text{th}}$ of its initial volume. The ratio of final pressure and initial pressure is:
($\gamma$ is the ratio of specific heats of the gas at constant pressure and at constant volume)
- The value closest to the thermal velocity of a Helium atom at room temperature (300 K) in $ms^{-1}$ is : [$k_B = 1.4 \times 10^{-23}$ J/K; $m_{He} = 7 \times 10^{-27}$ kg]
Important Questions from Heat and Thermodynamics
During the melting of a slab of ice at $273 \ K$ at atmospheric pressure:
- $\gamma_A$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. $\gamma_B$ is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If $\frac{\gamma_A}{\gamma_B} = \left(1 + \frac{1}{n}\right)$, then the value of n is _________.
- Match List - I with List - II.$\Delta Q$=Heat supplied
List - I List - II (A) Isobaric (I) $\Delta Q = \Delta W$ (B) Isochoric (II) $\Delta Q = \Delta U$ (C) Adiabatic (III) $\Delta Q = \text{zero}$ (D) Isothermal (IV) $\Delta Q = \Delta U + P\Delta V$
$\Delta W$ = Work done by the system
$\Delta U$=Change in internal energy
P = Pressure of the system
$\Delta V$ = Change in volume of the system
Choose the correct answer from the options given below : - There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K. If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K, at steady state the pressure in the vessels will be (in kPa).
Match the LIST-I with LIST-II Choose the correct answer from the options given below:
