Question

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

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

Step-by-step solution

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: \(Volume ≈ 15.03 \ \text{mL}\)

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: \(15.03 = πr^2(1.5)\)

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: \(r ≈ 0.98 \ \text{cm}\)

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.

Key concepts

Cylinder Volume Calculation

The volume of a cylinder is a crucial concept used in many scientific fields, including chemistry and physics. To find the volume of a cylinder, we use the formula \(V = \pi r^2 h\), where \(V\) is the volume, \(r\) is the radius of the cylinder's base, and \(h\) is its height.

In this problem, we find the volume based on the ethanol-filled tube. First, given the mass and density of ethanol, the volume is calculated. We learned that the volume of ethanol needed to fill the tube is 15.03 mL.

Then, we use this volume to set up the equation for the cylinder. The radius \(r\) is found by rearranging the formula to solve for \(r\) once the volume \(V\) and the height \(h\) (converted to the same unit) are known. After some calculations, we find the radius, and from that, the diameter. Knowing how to rearrange and solve geometry formulas like these is essential in many practical applications.

This exercise reinforces the understanding of cylindrical volume calculations by connecting geometric concepts with real-world measurements found in chemistry.

Ethanol Physical Properties

Ethanol, also known as ethyl alcohol, is a common organic compound with several uses in chemical and industrial applications. One of its critical properties is its density, which is 0.789 g/mL under standard conditions. This density value helps us convert between mass and volume, a necessary step in various calculations.

Because ethanol has a lower density than water (water's density is 1 g/mL), it is lighter for the same volume. This property is particularly important when considering ethanol's use in laboratory and industry settings.

Its chemical formula, \(C_2H_5OH\), signifies it as a volatile, flammable liquid with a noticeable alcohol smell. Ethanol's physical properties, including density, play a significant role when we aim to measure and calculate precise chemical reactions and processes in solutions and mixtures.

Measurement Conversion in Chemistry

Measurement conversion is a fundamental skill in chemistry as it allows scientists to express quantities in the most useful form. A common conversion in chemistry and physics involves transforming lengths, masses, and volumes into compatible units for calculations.

• In this exercise, the height of the cylindrical tube is given in centimeters, but we must ensure all measurements are compatible, with volume expressed in mL.
• Converting centimeters to milliliters might sound unusual, but it helps maintain consistent units when using the volume equation \(V = \pi r^2 h\).
• This conversion process ensures that computations are accurate and avoids errors due to unit mismatches.

Learning to convert measurements, such as from grams to milliliters using a substance's density, is critical for success in chemistry. It enables clear communication of results and ensures that calculations are performed correctly, providing a standard way of understanding and sharing chemical data internationally.