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Chapter 1: Problem 77

A \(15.0-\mathrm{cm}\) long cylindrical glass tube, sealed at one end, is filled with ethanol. The mass of ethanol needed to fill the tube is found to be \(11.86 \mathrm{~g} .\) The density of ethanol is \(0.789 \mathrm{~g} / \mathrm{mL}\). Calculate the inner diameter of the tube in centimeters.

Short Answer

Expert verified
The inner diameter of the cylindrical glass tube is approximately \(1.96 \ \text{cm}\).

Step by step solution

01

Calculate the volume of ethanol

To calculate the volume of ethanol, we need to use the given mass and density. The formula to find the volume with mass and density is: \(Volume = \frac{Mass}{Density}\) Given mass of ethanol = 11.86 g Density of ethanol = 0.789 g/mL Plug these values into the equation: \(Volume = \frac{11.86}{0.789}\) Calculate the volume: \(Volume ≈ 15.03 \ \text{mL}\)
02

Calculate the volume of the cylinder

The volume of a cylinder can be calculated using the formula: \(V = πr^2h\) where V is the volume, r is the radius, and h is the height of the cylinder. In this exercise, we know the volume of ethanol in the glass tube, which is 15.03 mL, and the height of the cylinder, given as 15 cm. We'll convert this height to mL by dividing it by 10, making it 1.5 mL. Now we can write the equation as follows: \(15.03 = πr^2(1.5)\)
03

Solve for the radius

Now, to find the radius, we will re-write the equation to isolate r: \(r^2 = \frac{15.03}{π(1.5)}\) Calculate the radius: \(r ≈ 0.98 \ \text{cm}\)
04

Calculate the diameter

Now that we have found the radius, we can easily find the diameter by multiplying it by 2: \(Diameter = 2 * radius\) Plug the value of the radius: \(Diameter ≈ 2 * 0.98\) Calculate the diameter: \(Diameter ≈ 1.96 \ \text{cm}\) So, the inner diameter of the cylindrical glass tube is approximately 1.96 cm.

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Most popular questions from this chapter

What exponential notation do the following abbreviations (e) \(\mathrm{M},(\mathrm{f}) \mathrm{k},(\mathrm{g}) \mathrm{n},(\mathrm{h}) \mathrm{m}\) represent: (a) \(\mathrm{d},(\mathbf{b}) \mathrm{c},(\mathrm{c}) \mathrm{f},(\mathrm{d}) \mu\) (i) p?

Give the derived SI units for each of the following quantities in base SI units: (a) acceleration = distance/time \(^{2},\) (b) force \(=\) mass \(\times\) acceleration, \((\mathrm{c})\) work \(=\) force \(\times\) distance, (d) \(\quad\) pressure \(=\) force/area, (e) \(\quad\) power \(=\) work/time, (f) velocity \(=\) distance/time, \((\mathrm{g})\) energy \(=\operatorname{mass} \times(\text { velocity })^{2}\).

By using estimation techniques, determine which of the following is the heaviest and which is the lightest: a 5-lb bag of potatoes, a \(5-\mathrm{kg}\) bag of sugar, or 1 gal of water \((\) density \(=1.0 \mathrm{~g} / \mathrm{mL})\)

(a) After the label fell off a bottle containing a clear liquid believed to be benzene, a chemist measured the density of the liquid to verify its identity. A 25.0 -mL portion of the liquid had a mass of \(21.95 \mathrm{~g}\). A chemistry handbook lists the density of benzene at \(15^{\circ} \mathrm{C}\) as \(0.8787 \mathrm{~g} / \mathrm{mL}\). Is the calculated density in agreement with the tabulated value? (b) An experiment requires \(15.0 \mathrm{~g}\) of cyclohexane, whose density at \(25^{\circ} \mathrm{C}\) is \(0.7781 \mathrm{~g} / \mathrm{mL}\). What volume of cyclohexane should be used? (c) A spherical ball of lead has a diameter of \(5.0 \mathrm{~cm}\). What is the mass of the sphere if lead has a density of \(11.34 \mathrm{~g} / \mathrm{cm}^{3}\) ? (The volume of a sphere is \((4 / 3) \pi r^{3}\) where \(r\) is the radius.)

(a) A cube of osmium metal \(1.500 \mathrm{~cm}\) on a side has a mass of \(76.31 \mathrm{~g}\) at \(25^{\circ} \mathrm{C}\). What is its density in \(\mathrm{g} / \mathrm{cm}^{3}\) at this temperature? (b) The density of titanium metal is \(4.51 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). What mass of titanium displaces \(125.0 \mathrm{~mL}\) of water at \(25^{\circ} \mathrm{C} ?\) (c) The density of benzene at \(15^{\circ} \mathrm{C}\) is \(0.8787 \mathrm{~g} / \mathrm{mL} .\) Calculate the mass of \(0.1500 \mathrm{~L}\) of benzene at this temperature.
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