Question

You have 750 g of water at 10.0\(^\circ\)C in a large insulated beaker. How much boiling water at 100.0\(^\circ\)C must you add to this beaker so that the final temperature of the mixture will be 75\(^\circ\)C?

Short answer

You need 1950 g of boiling water to reach 75°C.

Step-by-step solution

Identify the known quantities

You know the masses and initial temperatures of both the cold and hot water. The cold water mass is 750 g at 10.0°C, and the hot water is at 100.0°C. The final temperature you want is 75°C.

Use the principle of conservation of energy

The heat lost by the hot water will equal the heat gained by the cold water. Express this condition as: \( m_\text{hot} \cdot c \cdot (T_\text{initial, hot} - T_\text{final}) = m_\text{cold} \cdot c \cdot (T_\text{final} - T_\text{initial, cold}) \), where \( c \) is the specific heat capacity of water.

Substitute known values into the equation

The specific heat capacity cancels out. Substitute the masses and temperatures: \( m_\text{hot} \cdot (100 - 75) = 750 \cdot (75 - 10) \).

Solve for the unknown mass of the hot water

The equation becomes \( m_\text{hot} \cdot 25 = 750 \cdot 65 \). Calculate the product on the right side and then solve for \( m_\text{hot} \): \( m_\text{hot} = \frac{750 \times 65}{25} \).

Calculate the final answer

Perform the calculation \( m_\text{hot} = \frac{48750}{25} = 1950 \) g. Therefore, 1950 grams of boiling water is needed.

Key concepts

Conservation of Energy

In this exercise, we see an essential principle known as the conservation of energy in action. This principle states that energy cannot be created or destroyed but only transferred or converted from one form to another. In the context of heat transfer, this means the heat lost by one substance must equal the heat gained by another.
When you mix hot and cold water, the hot water loses thermal energy or heat, which is gained by the cold water. This process continues until both reach the same temperature, achieving a state called thermal equilibrium.
It’s important to remember that during this exchange, the total energy in the system remains constant. Only its form and distribution change among the water masses. By setting up an equation that reflects this constant energy, we can figure out unknown variables like the mass of water needed to achieve a specific temperature.

Specific Heat Capacity

The specific heat capacity ( \( c \) ) is a property of a material that defines how much heat energy it requires to change its temperature. For water, this value is approximately 4.18 Joules per gram per degree Celsius, but in this exercise, it cancels out due to both masses being in water.
This metric becomes especially useful in problems involving temperature changes because it allows us to calculate how much energy is needed to change the temperature of a given mass.

• It is constant for a given material, influencing how quickly it heats up or cools down.
• It tells us that more energy is needed to raise the temperature of a substance with a high specific heat capacity than one with a lower value.

In our exercise, even though we don't directly use the value of water's specific heat capacity, understanding it helps us realize why and how the energy exchange takes place during heating.

Thermal Equilibrium

Thermal equilibrium is reached when two substances in thermal contact no longer transfer heat between each other, meaning they have achieved the same temperature. In our exercise, after adding 1950 grams of boiling water to the colder water, both eventually reach the desired temperature of 75°C.
This concept is vital in determining the end condition of any heat transfer scenario. At equilibrium:

• The temperature of all involved bodies is the same.
• No more heat flows between them.

In practical applications, understanding thermal equilibrium is essential for solving problems involving mixed temperatures, such as in cooking or climate control systems. Identifying this stable point allows for calculating how much energy or mass is needed to achieve a desired temperature.