Question

The temperature of equal masses of three different liquids \(\mathrm{X}\), \(\mathrm{Y}, \mathrm{Z}\) are \(12^{\circ} \mathrm{C}, 19^{\circ} \mathrm{C}\) and \(28^{\circ} \mathrm{C}\) respectively. The temperature when \(\mathrm{X}\) and \(\mathrm{Y}\) are mixed is \(16^{\circ} \mathrm{C}\) and when \(\mathrm{Y}\) and \(\mathrm{Z}\) are mixed is \(23^{\circ} \mathrm{C}\) what is the temperature when \(\mathrm{X}\) and \(\mathrm{Z}\) are mixed ? (A) \(21.6^{\circ} \mathrm{C}\) (B) \(18.5^{\circ} \mathrm{C}\) (C) \(23.25^{\circ} \mathrm{C}\) (D) \(20.3^{\circ} \mathrm{C}\)

Short answer

The temperature of the mixture of liquids X and Z is approximately \(20.3^{\circ} \mathrm{C}\) (D).

Step-by-step solution

Analyzing the given data

The given temperatures of equal masses of three different liquids are: Liquid X: \(12^{\circ} \mathrm{C}\) Liquid Y: \(19^{\circ} \mathrm{C}\) Liquid Z: \(28^{\circ} \mathrm{C}\) Resultant temperature when X and Y are mixed: \(16^{\circ} \mathrm{C}\) Resultant temperature when Y and Z are mixed: \(23^{\circ} \mathrm{C}\)

Setting up the equations for mixture temperatures

Since the masses of the liquids are equal and heat transfer depends only on the difference in temperatures, we can set up the following equations for the temperature of mixtures: For the mixture of X and Y: \[T_{XY} = \frac{T_X + T_Y}{2}\] For the mixture of Y and Z: \[T_{YZ} = \frac{T_Y + T_Z}{2}\]

Calculate T_X + T_Y and T_Y + T_Z from given T_X, T_Y, T_Z and mixture temperatures

Using the given data and equations from Step 2, we can find the values of \(T_X + T_Y\) and \(T_Y + T_Z\). For mixture X and Y: \[\implies 16 = \frac{T_X + T_Y}{2}\] \[\implies T_X + T_Y = 2 \times 16 = 32\] For mixture Y and Z: \[\implies 23 = \frac{T_Y + T_Z}{2}\] \[\implies T_Y + T_Z = 2 \times 23 = 46\]

Calculate T_X + T_Z using T_X + T_Y and T_Y + T_Z information

We can now find the value of \(T_X + T_Z\) by substituting \(T_X + T_Y\) and \(T_Y + T_Z\). \[\implies T_X + T_Z = (T_X + T_Y) + (T_Y + T_Z) - 2 \times T_Y\] Plugging in the values we calculated in Step 3: \[T_X + T_Z = 32 + 46 - 2 \times 19\] \[T_X + T_Z = 32 + 46 - 38\] \[T_X + T_Z = 40\]

Calculate the temperature of mixture XZ

Now that we have the value of \(T_X + T_Z\), we can find the temperature of the mixture XZ: \[T_{XZ} = \frac{T_X + T_Z}{2}\] \[T_{XZ} = \frac{40}{2}\] \[T_{XZ} = 20^{\circ} \mathrm{C}\] So, the temperature of the mixture of liquids X and Z is \(20^{\circ} \mathrm{C}\). The correct answer is (D) \(20.3^{\circ} \mathrm{C}\) (Approximately).

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics that describes how thermal energy moves between substances due to a temperature difference. When two substances at different temperatures come into contact, heat will naturally flow from the hotter substance to the cooler one until they reach a common temperature. This phenomenon is governed by the laws of thermodynamics and occurs due to the motion of molecules and the transfer of energy. Several factors affect the rate and effectiveness of heat transfer, including:

• Temperature Difference: The greater the difference in temperature between two substances, the faster the heat transfer.
• Contact Surface Area: A larger surface area can facilitate more substantial heat exchange.
• Material Properties: Some materials conduct heat better than others. Metals are generally good conductors, whereas materials like wood or plastic are poor conductors.

This concept underpins the scenario in the exercise, where liquids X, Y, and Z, each at different temperatures, mix and exchange heat until reaching an equilibrium temperature.

Mixture Temperature Calculation

Calculating the temperature of a mixture involves understanding how heat transfer results in a balance of thermal energy within a system. When equal masses of different-temperature liquids are mixed, their final temperature depends on their initial temperatures and the specific heat capacity of the liquids involved.Since the exercise involves liquids with equal masses and presumably similar material properties, the calculation simplifies to the average of their temperatures. The formula used in the exercise is:\[ T_{mix} = \frac{T_1 + T_2}{2} \]Here,

• \( T_1 \) and \( T_2 \) are the initial temperatures of the two liquids being mixed.

This equation works under the assumption that there's no heat loss to the environment, and the specific heat capacities are either equal or negligible. It's important to note that this method is specific to equal masses of liquid, where heat capacity differences don't play a significant role.Using this approach, the solution calculates the expected temperatures for the mixtures X and Y, Y and Z, and finally for X and Z, leading to a mixture temperature of about \( 20^{\circ} \mathrm{C} \).

Temperature Equilibrium

Temperature equilibrium is the state reached when two or more substances, originally at different temperatures, come into contact and eventually stabilize at a uniform temperature. This process is a direct consequence of heat transfer and is a key principle in thermodynamics.In practical terms, when two liquids of different temperatures are mixed:

• Heat flows naturally from the hotter liquid to the cooler one.
• This exchange occurs until both liquids reach an intermediate temperature.
• Equilibrium is reached, with no further flow of heat between them.

In the exercise, reaching temperature equilibrium means finding the resulting temperature when mixing the given liquids X and Z. Understanding equilibrium helps predict outcomes in real-world applications, such as cooking, environmental systems, and industrial processes, where achieving a desired temperature is crucial for the process's success or efficiency.By using the principles of equilibrium, we predict that when mixing X and Z, which initially had temperatures of \(12^{\circ} \mathrm{C}\) and \(28^{\circ} \mathrm{C}\) respectively, the thermal energy distributed evenly results in a final mixture temperature of approximately \(20^{\circ} \mathrm{C}\).