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A material has mass density \(\rho,\) volume \(V\), and specific heat \(c .\) Which of the following is a correct expression for the heat exchange that occurs when the material's temperature changes by \(\Delta T\) in degrees Celsius? a) \((\rho c / V) \Delta T\) b) \((\rho c V)(\Delta T+273.15)\) c) \((\rho c V) / \Delta T\) d) \(\rho c V \Delta T\)

Short Answer

Expert verified
a) (蟻c / V) 螖T b) (蟻cV)(螖T+273.15) c) (蟻cV) / 螖T d) 蟻 c V 螖T Answer: d) 蟻 c V 螖T

Step by step solution

01

Write down the equation for heat exchange

The formula for calculating the heat exchange, \(Q\), when a material's temperature changes by \(\Delta T\) is: \(Q = mc\Delta T\)
02

Replace mass with mass density and volume

Since mass \(m\) can be written as the product of mass density (\(\rho\)) and volume (\(V\)): \(m = \rho V\) Now substitute this into the heat exchange equation from Step 1: \(Q = (\rho V)c\Delta T\) Now, we need to find the option that matches the above expression.
03

Compare with the given options

Let's look at each option: a) \((\rho c / V) \Delta T\) - This option divides \(\rho c\) by \(V\), which is not the same as our obtained expression. b) \((\rho c V)(\Delta T+273.15)\) - This option adds 273.15 to the temperature change \(\Delta T\). This is not in our obtained expression. c) \((\rho c V) / \Delta T\) - This divides \(\rho c V\) by \(\Delta T\), while our expression has \(\rho V c \Delta T\). d) \(\rho c V \Delta T\) - This option matches our obtained expression, which is the correct formula for the heat exchange when the temperature changes by \(\Delta T\). So the correct answer is: d) \(\rho c V \Delta T\)

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Key Concepts

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

Heat Exchange
Heat exchange is a fundamental concept in thermodynamics. It describes how energy moves as heat from one system or object to another. Heat exchange occurs during processes like heat conduction, convection, and radiation. When objects come into contact, or when temperature gradients exist, heat will flow from the hotter object or region to the cooler one.
Understanding heat exchange is crucial for calculating energy transfers during heating and cooling.
- **Formula**: The formula for heat exchanged during a temperature change is given by: \[ Q = mc\Delta T \] where: - \( Q \): Heat exchanged (often in Joules) - \( m \): Mass of the object (in kilograms) - \( c \): Specific heat capacity (J/kg鈩) - \( \Delta T \): Change in temperature (in 鈩)The idea is to know how much heat is absorbed or released by the object to understand energy balance in systems.
Temperature Change
Temperature change (\( \Delta T \)) measures how the heat exchange impacts the thermal state of a material. It highlights shifts in thermal energy, quantified by a temperature increase or decrease.
When a substance gains heat, its temperature rises. Conversely, when it loses heat, the temperature drops.
- **Role in Calculations**: The change in temperature is crucial for calculating heat exchange because it helps determine how much energy is required to achieve a certain thermal state. Given by: \[ \Delta T = T_{final} - T_{initial} \] where: - \( T_{final} \): Final temperature after heat exchange - \( T_{initial} \): Initial temperature before heat exchange Understanding temperature changes helps predict how substances respond to energy inputs, crucial for designing thermal systems.
Mass Density
Mass density (\( \rho \)), often represented as \( \rho \), is the mass per unit volume of a material. It's expressed in units like kg/m鲁 and is a measure of how tightly matter is packed in a substance.
Mass density links the material's mass to its volume and is a vital factor in calculating the mass when volume and mass density are known.- **Relation to Heat Exchange**: During heat exchange calculations, mass density helps convert volume into mass since: \[ m = \rho V \] where: - \( m \): Mass (kg) - \( \rho \): Mass density (kg/m鲁) - \( V \): Volume (m鲁)Knowing the mass density allows you to understand the physical properties of a material, impacting how it stores and transfers energy.
Volume
Volume (\( V \)) is the space that a substance occupies, measured in cubic meters (m鲁) or liters in metric units. Volume is crucial when dealing with fluids and gases, and it affects calculations in physics and engineering.
When talking about solids, liquids, or gases, volume helps us determine the space a given mass of matter will occupy. In many thermal calculations, knowing the volume is key to linking with other properties like mass density.- **Part of Heat Exchange Formula**: In the heat exchange formula, volume helps connect the physical dimensions of a material with its heat capacity. - By knowing the volume and mass density: \[ m = \rho V \] we can calculate how much energy in heat has been exchanged when the temperature changes. Volume is essential for understanding and managing physical and chemical processes, as it influences the behavior of substances when heated or cooled.

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Most popular questions from this chapter

A metal brick found in an excavation was sent to a testing lab for nondestructive identification. The lab weighed the sample brick and found its mass to be \(3.0 \mathrm{~kg} .\) The brick was heated to a temperature of \(3.0 \cdot 10^{2}{ }^{\circ} \mathrm{C}\) and dropped into an insulated copper calorimeter of mass 1.5 kg containing \(2.0 \mathrm{~kg}\) of water at \(2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C} .\) The final temperature at equilibrium was noted to be \(31.7^{\circ} \mathrm{C}\). By calculating the specific heat of the sample from this data, can you identify the brick's material?

The internal energy of a gas is \(500 .\) J. The gas is compressed adiabatically, and its volume decreases by \(100 . \mathrm{cm}^{3} .\) If the pressure applied on the gas during compression is \(3.00 \mathrm{~atm},\) what is the internal energy of the gas after the adiabatic compression?

An air-cooled motorcycle engine loses a significant amount of heat through thermal radiation according to the Stefan-Boltzmann equation. Assume that the ambient temperature is \(T_{0}=27^{\circ} \mathrm{C}(300 \mathrm{~K})\). Suppose the engine generates 15 hp \((11 \mathrm{~kW})\) of power and, due to several deep surface fins, has a surface area of \(A=0.50 \mathrm{~m}^{2}\). A shiny engine has an emissivity \(e=0.050\), whereas an engine that is painted black has \(e=0.95 .\) Determine the equilibrium temperatures for the black engine and the shiny engine. (Assume that radiation is the only mode by which heat is dissipated from the engine.)

Knife blades are often made of hardened carbon steel. The hardening process is a heat treatment in which the blade is first heated to a temperature of \(1346^{\circ} \mathrm{F}\) and then cooled down rapidly by immersing it in a bath of water. To achieve the desired hardness, after heating to \(1346^{\circ} \mathrm{F}\), a blade needs to be brought to a temperature below \(5.00 \cdot 10^{2}{ }^{\circ} \mathrm{F}\). If the blade has a mass of \(0.500 \mathrm{~kg}\) and the water is in an open copper container of mass \(2.000 \mathrm{~kg}\) and sufficiently large volume, what is the minimum quantity of water that needs to be in the container for this hardening process to be successful? Assume the blade is not in direct mechanical (and thus thermal) contact with the container, and neglect cooling through radiation into the air. Assume no water boils but reaches \(100^{\circ} \mathrm{C} .\) The heat capacity of copper around room temperature is \(c_{\text {copper }}=386 \mathrm{~J} /(\mathrm{kg} \mathrm{K}) .\) Use the data in the table below for the heat capacity of carbon steel

How would the rate of heat transfer between a thermal reservoir at a higher temperature and one at a lower temperature differ if the reservoirs were in contact with a 10 -cm-long glass rod instead of a 10 -m-long aluminum rod having an identical cross-sectional area?

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