Question

A small electric immersion heater is used to boil \(136 \mathrm{~g}\) of water for a cup of instant coffee. The heater is labeled 220 watts. Calculate the time required to bring this water from \(23.5^{\circ} \mathrm{C}\) to the boiling point, ignoring any heat losses.

Short answer

After performing all the necessary calculations, insert the final result for the time (in minutes) here.

Step-by-step solution

Calculate the Heat Required

In order to calculate the heat required to heat the water, we need to know the mass of the water, the specific heat of the water, and the temperature change. We know the mass of the water is \(136 \mathrm{~g}\), the specific heat of the water is \(4.184 \mathrm{~J/g^{\circ}C}\), and the change in temperature is the difference between the final temperature (the boiling point of water, \(100 ^{\circ} \mathrm{C}\)), and the initial temperature (\(23.5 ^{\circ} \mathrm{C}\)). We then insert these values into the formula \(Q = mc\Delta T\) to calculate the heat required. The calculation will be: \(Q = (136 \mathrm{~g}) \cdot (4.184 \mathrm{~J/g^{\circ}C}) \cdot [(100 - 23.5) ^{\circ} \mathrm{C}]\)

Calculate Time

After calculating the heat required, the next step is to determine the time needed to deliver this amount of heat. We know that power, which is energy delivered per unit time, is given as 220 watts. Therefore, by using the formula \(t = Q/P\), we can calculate the time required. Insert the calculated value for Q and the given value for P into this equation to calculate the time.

Convert Units

Since the power was given in watts, which is equivalent to joules per second, the time calculated in step 2 will be in seconds. To convert this to more convenient units (minutes), divide by 60.

Key concepts

Specific Heat Capacity

When heating a substance, specific heat capacity is a crucial factor. It's a measure of how much energy is needed to raise the temperature of a given mass of a substance by one degree Celsius. For water, this value is quite high, at 4.184 J/g°C, meaning it takes a good amount of energy to change its temperature.

• Specific heat capacity depends on the material, which is why different substances heat up at different rates.
• It's commonly used in calculations involving energy changes in heating and cooling processes.
• To find the heat required, use the formula: Q = mcΔT, where m is mass, c is specific heat capacity, and ΔT is the temperature change.

Understanding specific heat helps explain why large bodies of water like oceans can absorb and store huge amounts of energy.

Energy Calculations

Energy calculations are fundamental to understanding heat transfer applications, like heating water. The energy supplied, or the work done, is often calculated using the formula Q = mcΔT, where each variable needs proper units.

• For water, you'll use specific heat capacity in J/g°C, mass in grams, and temperature change in degrees Celsius.
• These calculations allow you to predict how much energy you'll need for heating, and plan accordingly.

Energy calculations aren't just academic; they help us design efficient systems for heating, ensure that electrical appliances are used safely and effectively, and understand energy requirements in various applications.

Electric Power

Electric power defines how quickly energy is used or transferred, and in electrical circuits, it's measured in watts. This is equivalent to the energy used per unit time. For the immersion heater in the exercise:

• Power output is given as 220 watts, which means it transfers 220 joules of energy every second.
• The formula P = Q/t (relating power, energy and time) is crucial for time-related energy calculations.

In practical terms, electric power helps determine the efficiency and speed at which electrical devices operate, which is essential for devices like heaters where timed operations are crucial.

Boiling Point

The boiling point of a substance is the temperature at which a liquid turns into vapor. For water, this is typically 100°C at standard atmospheric pressure.

• Atmospheric pressure significantly affects boiling point; lower pressure (like at high altitudes) lowers the boiling point.
• Knowing the boiling point is important for calculating energy needs, as it sets the final temperature in heating tasks.

Why do we care? Understanding boiling points allows us to efficiently heat substances to their boiling state, critical in cooking, chemical reactions, and industrial processes.

Temperature Change

Temperature change represents how much a substance's temperature shifts. It's the difference between the final and initial temperatures, represented by ΔT in calculations.

• For the exercise, the initial water temperature is 23.5°C, and the final temperature is 100°C, giving a temperature change of 76.5°C.
• Accurate measurement and calculation of temperature change are vital for determining the energy needed to reach desired temperatures.

Understanding temperature change is key to problem-solving in physics and engineering, allowing for precise control and manipulation of heat energy.