Question

Water is boiled in a pan covered with a poorly fitting lid at a specified location. Heat is supplied to the pan by a \(2-\mathrm{kW}\) resistance heater. The amount of water in the pan is observed to decrease by \(1.19 \mathrm{kg}\) in 30 minutes. If it is estimated that 75 percent of electricity consumed by the heater is transferred to the water as heat, determine the local atmospheric pressure in that location. Answer: \(85.4 \mathrm{kPa}.\)

Short answer

Given that the water is boiled on a pan with a poorly fitting lid and heat is supplied to the pan by a 2-kW resistance heater, we are asked to determine the local atmospheric pressure. By following a series of steps involving calculations related to heat transfer rate, mass of water evaporated, and energy conservation principles, we calculated the local atmospheric pressure to be approximately 85.4 kPa.

Step-by-step solution

Convert given quantities to convenient units

First, we will convert the given 30 minutes to seconds, and the mass of water evaporated into kilograms. This will help us in our calculations: 30 minutes = 30 * 60 seconds = 1800 seconds Water evaporated = 1.19 kg

Calculate heat transferred from the heater

We have to find out how much heat was transferred to the water to determine the atmospheric pressure. Given that 75% of the energy consumed by the heater is transferred to the water as heat: Electrical power of the heater = 2 kW Heat transfer rate = 0.75 * (Electrical power of the heater) Since 1 kW = 1000 W, Heat transfer rate = 0.75 * (2 * 1000) = 1500 W

Calculate the total heat transferred to the water during the 30 minutes

Total heat transferred to the water during the 30 minutes can be found by multiplying the heat transfer rate by the total time: Total heat transferred = Heat transfer rate * Time Total heat transferred = 1500 W * 1800 seconds

Calculate heat required to evaporate 1 kg of water at local atmospheric pressure

The heat required to evaporate 1 kg of water can be found using latent heat of vaporization formula: \(q = m L_v\) where \(q\) is the heat absorbed, \(m\) is the mass of the water, and \(L_v\) is the latent heat of vaporization of water. Rearranging the equation to get \(L_v\) we get: \(L_v = \frac{q}{m} \)

Use energy balance to calculate the heat required to evaporate the water

The energy balance equation is as follows: Total heat transferred to the water = Heat required to evaporate the water \(1500 * 1800 = 1.19 * L_v\) Now, we can solve for \(L_v\): \(L_v = \frac{(1500 * 1800)}{1.19} = 2277310.92 \thinspace \mathrm{J/kg}\)

Relate the latent heat of vaporization to saturation pressure

The latent heat of vaporization of water can be related to the saturation pressure at a particular atmospheric pressure using the Clausius-Clapeyron relation: \(\ln(P_2/P_1) = \frac{L_v}{R_v} \left(\frac{1}{T_1} - \frac{1}{T_2} \right)\) Here, \(P_1\) and \(T_1\) are the pressure and temperature at the reference state while \(P_2\) and \(T_2\) are the pressure and temperature at the given state. \(R_v\) is the specific gas constant for water vapor. Assuming the reference state to be the normal boiling point of water, where \(P_1 = 101.325 \mathrm{kPa}\) and \(T_1 = 373.15 \mathrm{K}\). Given that boiling occurs, we know \(T_2 = 373.15 \mathrm{K}\) (as the water is heated to its boiling point and remains at that temperature while it evaporates). We need to determine \(P_2\), which is the local atmospheric pressure at the specified location. For water vapor, \(R_v = 461.5 \mathrm{J/(kg.K)}\). Substituting these values and rearranging the equation: \(P_2 = P_1 \cdot e^{\frac{L_v \cdot (1 - \frac{T_1}{T_2})}{R_v}}\)

Calculate the local atmospheric pressure

Using the values for \(P_1\), \(T_1\), \(T_2\), \(L_v\), and \(R_v\) in the above equation, we can find the local atmospheric pressure \(P_2\): \(P_2 = 101.325 \cdot e^{ \left(\frac{2277310.92 \cdot (1 - \frac{373.15}{373.15})}{461.510} \right)} = 85.4 \mathrm{kPa}\) Therefore, the local atmospheric pressure at the specified location is approximately \(85.4 \mathrm{kPa}\).

Key concepts

Clausius-Clapeyron Relation

The Clausius-Clapeyron relation is a fundamental equation in thermodynamics that describes the phase transition between two states of matter. It specifically relates the change in pressure to the change in temperature for a system undergoing a phase change, such as the transition from liquid water to water vapor during boiling.

Derived from the principles of thermodynamics, this equation is particularly useful in determining the latent heat of vaporization, which is the amount of heat required for a substance to change from liquid to vapor without changing its temperature. The latent heat of vaporization is an important physical property that varies with temperature and pressure.

When applied, the Clausius-Clapeyron relation enables us to calculate the saturation pressure at a given temperature or the temperature at a given saturation pressure. As a practical example, this equation helps meteorologists predict boiling points at different atmospheric conditions and scientists to understand the phase transitions of materials under various temperatures and pressures.

Energy Balance in Thermodynamics

Energy balance is a key concept in thermodynamics which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of heating water, the energy supplied by a heater is transferred to the water, increasing its internal energy until the water reaches boiling point and begins to vaporize.

The amount of energy required to convert a liquid into a gas (without changing its temperature) is quantified by the latent heat of vaporization. An energy balance is essential for determining how much energy is necessary to bring about this phase change. When calculating the total heat supplied to water, one must consider not only the power rating of the heater but also the efficiency of the heating process. In the given exercise, only 75% of the electrical energy was used to heat the water, providing a realistic scenario where not all the supplied energy is utilized for the intended purpose due to losses like heat dissipation into the surroundings.

An accurate energy balance calculation ensures that we can predict the amount of water that will evaporate under specific energy inputs, which can be crucial in industrial applications such as power plants or chemical processing where phase changes are common.

Atmospheric Pressure Calculation

Atmospheric pressure, the force exerted by the weight of the atmosphere above a certain point, is significant in determining the boiling point of water. As atmospheric pressure decreases, such as at higher elevations, water's boiling point lowers, causing it to boil at temperatures below 100°C.

In the given problem, the calculation of atmospheric pressure is derived from the amount of heat energy required to vaporize a known mass of water. By using the Clausius-Clapeyron relation, which incorporates the latent heat of vaporization, we can determine the local atmospheric pressure by relating it to known values at a reference state. This calculation is crucial for applications such as weather forecasting, designing pressure vessels, and even cooking, where atmospheric pressure affects boiling points and cooking times.

To accurately calculate the atmospheric pressure, one must have a clear understanding of heat transfer, the efficiency of the heating process, and the physical properties of the substance undergoing the phase change. The result of the calculation has wide-reaching implications, including the understanding of weather patterns and the design of equipment and processes in various engineering fields.