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Chapter 18: Problem 8

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) (ρc / V) ΔT b) (ρcV)(ΔT+273.15) c) (ρcV) / ΔT d) ρ c V ΔT Answer: d) ρ c V ΔT

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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