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Chapter 21: Problem 34

An aluminum cup of \(110 \mathrm{~cm}^{3}\) capacity is filled with glycerin at \(22^{\circ} \mathrm{C}\). How much glycerin, if any, will spill out of the cup if the temperature of the cup and glycerin is raised to \(28^{\circ} \mathrm{C}\) ? (The coefficient of volume expansion of glycerin is \(5.1 \times\) \(\left.10^{-4} / \mathrm{C}^{\circ} .\right)\)

Short Answer

Expert verified
By the increase in temperature, the volume of glycerin increased by 3.366 cm^3. So, this amount of glycerin will spill out of the cup.

Step by step solution

01

Identify the knowns and unknowns

From the problem, the initial volume of glycerin V0 = 110 cm^3, β = 5.1x10^-4 /°C, initial temperature = 22°, final temperature = 28°C. Our aim is to calculate ∆V.
02

Calculate the change in temperature

Calculate the change in temperature using the formula ∆T = Final temperature - Initial temperature. Therefore we get ∆T = 28°C - 22°C = 6°C.
03

Calculate the change in volume

Substitute the values of β, V0 and ∆T in the formula for volume expansion ∆V = βV0∆T. Therefore we get ∆V = 5.1x10^-4 /°C x 110 cm^3 x 6°C = 3.366 cm^3.

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Most popular questions from this chapter

An air bubble of \(19.4 \mathrm{~cm}^{3}\) volume is at the bottom of a lake \(41.5 \mathrm{~m}\) deep where the temperature is \(3.80^{\circ} \mathrm{C}\). The bubble rises to the surface, which is at a temperature of \(22.6^{\circ} \mathrm{C}\). Take the temperature of the bubble to be the same as that of the surrounding water and find its volume just before it reaches the surface.

A thermocouple is formed from two different metals, joined at two points in such a way that a small voltage is produced when the two junctions are at different temperatures. In a particular iron-constantan thermocouple, with one junction held at \(0^{\circ} \mathrm{C}\), the output voltage varies linearly from 0 to \(28.0 \mathrm{mV}\) as the temperature of the other junction is raised from 0 to \(510^{\circ} \mathrm{C}\). Find the temperature of the variable junction when the thermocouple output is \(10.2 \mathrm{mV}\).

( \(a\) ) Prove that the change in period \(P\) of a physical pendulum with temperature is given by \(\Delta P=\frac{1}{2} \alpha P \Delta T .(b)\) A clock pendulum made of invar has a period of \(0.500 \mathrm{~s}\) and is accurate at \(20^{\circ} \mathrm{C}\). If the clock is used in a climate where the temperature averages \(30^{\circ} \mathrm{C}\), what approximate correction to the time given by the clock is necessary at the end of 30 days?

A cylinder placed in frictionless bearings is set rotating about its axis. The cylinder is then heated, without mechanical contact, until its radius is increased by \(0.18 \%\). What is the percent change in the cylinder's ( \(a\) ) angular momentum, \((b)\) angular velocity, and ( \(c\) ) rotational energy?

(a) Using the ideal gas law and the definition of the coefficient of volume expansion (Eq. \(21-12\) ), show that \(\beta=1 / T\) for an ideal gas at constant pressure. \((b)\) In what units must \(T\) be expressed? If \(T\) is expressed in those units, can you express \(\beta\) in units of \(\left(\mathrm{C}^{\circ}\right)^{-1} ?(c)\) Estimate the value of \(\beta\) for an ideal gas at room temperature.
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