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
- 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. ($R = 8.314 \text{ J/mol.K}$)
A thermodynamic system is taken through the cyclic process ABC as shown in the figure. The total work done by the system during the cycle $ABC$ is _________ $\text{J}$.
- In 5 minutes, a body cools from $75^{\circ}C$ to $65^{\circ}C$ at room temperature of $25^{\circ}C$. The temperature of body at the end of next 5 minutes is __________ $^{\circ}C$.
- The area of cross-section of a railway track is $0.01 \text{ m}^2$. The temperature variation is $10^{\circ}C$. Coefficient of linear expansion of material of track is $10^{-5}/^{\circ}C$. The energy stored per meter in the track is __________ J/m.<br>(Young's modulus of material of track is $10^{11} \text{ Nm}^{-2}$)
- Two cylindrical rods A and B made of different materials, are joined in a straight line. The ratios of lengths, radii and thermal conductivites of these rods are: $\frac{L_A}{L_B} = 1$, $\frac{r_A}{r_B} = 2$ and $\frac{K_A}{K_B} = \frac{1}{2}$. The free ends of rods A and B are maintained at 400 K, 200 K, respectively. The temperature of rods interface is ________ K, when equilibrium is established.
The internal energy of air in $4 m \times 4 m \times 3 m$ sized room at 1 atmospheric pressure will be ________ $\times 10^6$ J.
(Consider air as diatomic molecule)- $\gamma_A$ 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 _________.
- A certain gas is isothermally compressed to $\left(\frac{1}{3}\right)^{\text{rd}}$ of its initial volume ($V_o = 3$ litre) by applying required pressure. If the bulk modulus of the gas is $3 \times 10^5 \text{ N/m}^2$, the magnitude of work done on the gas is _________ J.
- 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.
- 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.
Important Questions from Heat and Thermodynamics
- 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. ($R = 8.314 \text{ J/mol.K}$)
A thermodynamic system is taken through the cyclic process ABC as shown in the figure. The total work done by the system during the cycle $ABC$ is _________ $\text{J}$.
- Two identical metal bars are heated in two different temperatures and allowed to cool in the same surroundings. Which one of the following figures correctly shows their cooling curves?
- In 5 minutes, a body cools from $75^{\circ}C$ to $65^{\circ}C$ at room temperature of $25^{\circ}C$. The temperature of body at the end of next 5 minutes is __________ $^{\circ}C$.
- The area of cross-section of a railway track is $0.01 \text{ m}^2$. The temperature variation is $10^{\circ}C$. Coefficient of linear expansion of material of track is $10^{-5}/^{\circ}C$. The energy stored per meter in the track is __________ J/m.<br>(Young's modulus of material of track is $10^{11} \text{ Nm}^{-2}$)
