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Chapter 20: Problem 57

A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique of flow calorimetry involves measuring the temperature difference between the input and output points of a flowing stream of the liquid while energy is added by heat at a known rate. A liquid of density \(\rho\) flows through the calorimeter with volume flow rate \(R\). At steady state, a temperature difference \(\Delta T\) is established between the input and output points when energy is supplied at the rate \(\mathscr{P}\). What is the specific heat of the liquid?

Short Answer

Expert verified
The specific heat \(c\) of the liquid is given by the equation \(c = \frac{\mathscr{P}}{\rho R \Delta T}\).

Step by step solution

01

Express the heat supplied

First express the heat \(\mathscr{P}\) added to the system. This is given by the equation \(\mathscr{P} = mc\Delta T\). But here, the mass flow rate \(m\) is equal to the product of the liquid's density \(\rho\) and the volume flow rate \(R\). So, the equation becomes \(\mathscr{P} = \rho R c\Delta T\).
02

Solve for the specific heat

We are asked to find the specific heat \(c\) of the liquid. Rearrange the equation from step 1 to express \(c\) in terms of the other variables. This gives \(c = \frac{\mathscr{P}}{\rho R \Delta T}\).
03

Interpret the result

The equation \(c = \frac{\mathscr{P}}{\rho R \Delta T}\) gives the specific heat of the liquid in terms of the power supplied to the calorimeter, the liquid's density and volume flow rate, and the temperature difference across the calorimeter. It tells us that the specific heat is directly proportional to the power and inversely proportional to the product of the density, volume flow rate and temperature difference.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Flow Calorimetry
Flow calorimetry is a technique to measure the thermal properties of liquids by assessing how they absorb or dissipate heat as they flow through a calorimeter. When a liquid with a certain density \(\rho\) flows at a consistent volume flow rate \(R\), energy is added at a steady rate, leading to a change in temperature from the inlet to the outlet.

This method provides an accurate way of determining specific heat, which is a measure of how much energy is needed to change the temperature of a certain mass of a substance by one degree Celsius. In practice, steady-state conditions are important for accurate measurements, ensuring that the temperature difference and energy input are consistent over time.
Heat Transfer
Heat transfer in the context of flow calorimetry refers to the process by which thermal energy moves from the calorimeter to the liquid or vice versa. There are three primary modes of heat transfer: conduction, convection, and radiation. However, within the confines of a flow calorimeter, conduction and convection are the primary concerns, as these are the means through which energy is imparted to the flowing liquid. An efficient heat transfer process is critical to achieving a steady temperature difference that is necessary for accurate calorimetric measurements.
Temperature Difference
The temperature difference \(\Delta T\) in the context of flow calorimetry is the measurable change in temperature between two points, usually the input and output points of the liquid as it flows through the calorimeter. A consistent and measurable \(\Delta T\) signifies a steady state and is essential for calculating the specific heat of the liquid. Understanding the temperature difference allows scientists and engineers to deduce the quantity of thermal energy absorbed or released by the fluid during the process.
Volume Flow Rate
Volume flow rate \(R\) is the quantity of liquid passing through a point or surface per unit time in flow calorimetry. Commonly measured in cubic meters per second or liters per minute, the volume flow rate not only affects the heat transfer dynamics but is also a crucial variable in calculating the specific heat capacity of a liquid. It is important that the flow rate is maintained constant during the experiment to ensure that the energy input leads to a uniform temperature increase across the liquid's flow path.
Thermal Energy
Thermal energy, in the setting of flow calorimetry, refers to the internal energy present in the liquid as a result of its temperature. When a liquid flows through a calorimeter, thermal energy is either added or subtracted, leading to a change in temperature indicated by \(\Delta T\).

Understanding thermal energy's role is essential because the specific heat capacity is fundamentally about how much this thermal energy changes for a unit mass of substance per degree temperature change. The rate at which this energy is added or removed in flow calorimetry is represented by the power \(\mathscr{P}\) and is essential for calculating the specific heat \(c\).

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Most popular questions from this chapter

One mole of an ideal gas is contained in a cylinder with a movable piston. The initial pressure, volume, and temperature are \(P_{i}, V_{i},\) and \(T_{i},\) respectively. Find the work done on the gas for the following processes and show each process on a PV diagram: (a) An isobaric compression in which the final volume is half the initial volume. (b) An isothermal compression in which the final pressure is four times the initial pressure. (c) An isovolumetric process in which the final pressure is three times the initial pressure.

A \(1.50-\mathrm{kg}\) iron horseshoe initially at \(600^{\circ} \mathrm{C}\) is dropped into a bucket containing \(20.0 \mathrm{kg}\) of water at \(25.0^{\circ} \mathrm{C} .\) What is the final temperature? (Ignore the heat capacity of the container, and assume that a negligible amount of water boils away.)

A 670-kg meteorite happens to be composed of aluminum. When it is far from the Earth, its temperature is \(-15^{\circ} \mathrm{C}\) and it moves with a speed of \(14.0 \mathrm{km} / \mathrm{s}\) relative to the Earth. As it crashes into the planet, assume that the resulting additional internal energy is shared equally between the meteor and the planet, and that all of the material of the meteor rises momentarily to the same final temperature. Find this temperature. Assume that the specific heat of liquid and of gaseous aluminum is \(1170 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}.\)

Systematic use of solar energy can yield a large saving in the cost of winter space heating for a typical house in the north central United States. If the house has good insulation, you may model it as losing energy by heat steadily at the rate \(6000 \mathrm{W}\) on a day in April when the average exterior temperature is \(4^{\circ} \mathrm{C},\) and when the conventional heating system is not used at all. The passive solar energy collector can consist simply of very large windows in a room facing south. Sunlight shining in during the daytime is absorbed by the floor, interior walls, and objects in the room, raising their temperature to \(38^{\circ} \mathrm{C} .\) As the sun goes down, insulating draperies or shutters are closed over the windows. During the period between 5: 00 P.M. and 7: 00 A.M. the temperature of the house will drop, and a sufficiently large "thermal mass" is required to keep it from dropping too far. The thermal mass can be a large quantity of stone (with specific heat \(850 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) ) in the floor and the interior walls exposed to sunlight. What mass of stone is required if the temperature is not to drop below \(18^{\circ} \mathrm{C}\) overnight?

An aluminum rod \(0.500 \mathrm{m}\) in length and with a crosssectional area of \(2.50 \mathrm{cm}^{2}\) is inserted into a thermally insulated vessel containing liquid helium at \(4.20 \mathrm{K}\). The rod is initially at \(300 \mathrm{K}\). (a) If half of the rod is inserted into the helium, how many liters of helium boil off by the time the inserted half cools to \(4.20 \mathrm{K} ?\) (Assume the upper half does not yet cool.) (b) If the upper end of the rod is maintained at \(300 \mathrm{K},\) what is the approximate boil-off rate of liquid helium after the lower half has reached \(4.20 \mathrm{K} ?\) (Aluminum has thermal conductivity of \(31.0 \mathrm{J} / \mathrm{s} \cdot \mathrm{cm} \cdot \mathrm{K}\) at \(4.2 \mathrm{K} ;\) ignore its temperature variation. Aluminum has a specific heat of \(0.210 \mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C}\) and density of \(2.70 \mathrm{g} / \mathrm{cm}^{3} .\) The density of liquid helium is \(0.125 \mathrm{g} / \mathrm{cm}^{3} .\) )
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