Question

Popcorn pops because the horny endosperm, a tough, elastic material, resists gas pressure within the heated kernel until the pressure reaches explosive force. A 0.25 -mL kernel has a water content of \(1.6 \%\) by mass, and the water vapor reaches \(170^{\circ} \mathrm{C}\) and 9.0 atm before the kernel ruptures. Assume that water vapor can occupy \(75 \%\) of the kernel's volume. (a) What is the mass (in g) of the kernel? (b) How many milliliters would this amount of water vapor occupy at \(25^{\circ} \mathrm{C}\) and 1.00 atm?

Short answer

a) The mass of the kernel is 0.25 g. b) The water vapor would occupy 1.13 mL at 25°C and 1.00 atm.

Step-by-step solution

Calculate the mass of the water in the kernel

The kernel has a water content of 1.6% by mass. To find the mass of the water in the kernel, multiply the total volume of the kernel (0.25 mL) by the mass fraction of water (1.6 \%). Since the density of water is approximately 1 g/mL, the mass of the water can be found using the relation: \[ Mass_{water} = Volume_{kernel} \times Mass \, Fraction_{water} \]Thus, \[ Mass_{water} = 0.25 \, mL \times \frac{1.6}{100} \approx 0.004 \, g \]

Find the total mass of the kernel

Use the mass fraction to determine the total mass of the kernel. Rearrange the formula for the mass fraction: \[ Mass \, Fraction_{water} = \frac{Mass_{water}}{Mass_{kernel}} \]Solving for the mass of the kernel: \[ Mass_{kernel} = \frac{Mass_{water}}{Mass \, Fraction_{water}} = \frac{0.004 \, g}{0.016} = 0.25 \, g \]

Calculate the volume of water vapor at rupture conditions

Assuming that water vapor occupies 75% of the kernel's volume at rupture conditions, calculate this volume: \[ Volume_{vapor} = Volume_{kernel} \times 0.75 \]\[ Volume_{vapor} = 0.25 \, mL \times 0.75 = 0.1875 \, mL \]

Apply the Ideal Gas Law to find the moles of water vapor

Use the Ideal Gas Law: \[ PV = nRT \]Rearrange to solve for the number of moles (n): \[ n = \frac{PV}{RT} \]Given that pressure (P) is 9.0 atm, volume (V) is 0.1875 mL (convert to L: 0.0001875 L), gas constant (R) is 0.0821 L•atm/(K•mol), and temperature (T) is 170°C (convert to Kelvin: 443 K): \[ n = \frac{9.0 \, atm \times 0.0001875 \, L}{0.0821 \, (L \, atm)/(K \, mol) \times 443 \, K} \approx 4.63 \times 10^{-5} \, mol \]

Calculate the volume of water vapor at new conditions

Use the Ideal Gas Law again to calculate the volume at 25°C and 1.00 atm. Given that temperature (T) is 25°C (convert to Kelvin: 298 K), and pressure (P) is 1.00 atm, use the moles calculated previously: \[ V = \frac{nRT}{P} = \frac{4.63 \times 10^{-5} \, mol \times 0.0821 \, (L \, atm)/(K \, mol) \times 298 \, K}{1.00 \, atm} \approx 0.00113 \, L \]Convert this volume to milliliters: \[ 0.00113 \, L \times 1000 \, mL/L = 1.13 \, mL \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that helps us understand the behavior of gases. It is expressed as: \[ PV = nRT \] Here,

• P represents the pressure of the gas
• V stands for volume
• n is the number of moles of the gas
• R is the gas constant
• T denotes temperature in Kelvin

In our exercise, we use this law to calculate the number of moles of water vapor inside the popcorn kernel and later to find the volume of the vapor at different conditions. By rearranging the equation, you can solve for different variables depending on what you need to find, such as pressure, volume, number of moles, or temperature.

Endosperm

In a popcorn kernel, the endosperm is the starchy interior part that gives popcorn its characteristic pop. This endosperm is surrounded by a tough, elastic shell that can withstand significant amounts of pressure.
The endosperm has two types: the horny endosperm, which is hard and vitreous, and the floury endosperm, which is soft and starchy. When the kernel is heated, the water inside turns to steam, creating high pressure. The endosperm's role is to resist this pressure until it becomes too much, leading to an explosive rupture that pops the corn.

Water Vapor Volume Calculation

Calculating the volume of water vapor was done through several steps. Initially, we assumed that the water vapor occupies 75% of the kernel's volume at rupture conditions. This volume was divided among different conditions using the Ideal Gas Law.
First, we found the volume the water vapor would occupy inside the kernel just before it popped. To do this, we used the kernel's total volume and the given percentage to get: \[ Volume_{vapor} = Volume_{kernel} \times 0.75 \] Next, with this volume and the conditions at rupture (high temperature and pressure), we applied the Ideal Gas Law to calculate the moles of water vapor. Using these moles, we calculated the volume of water vapor at a new set of conditions (lower temperature and pressure).

Mass Fraction Calculation

The mass fraction gives the ratio of the mass of one component to the total mass. In our popcorn problem, we used it to find how much of the kernel's mass is water. First, we calculated the mass of the water in the kernel:
\[ Mass_{water} = Volume_{kernel} \times Mass \, Fraction_{water} \] Then we rearranged this to find the total mass of the kernel using the known mass fraction of water: \[ Mass_{kernel} = \frac{Mass_{water}}{Mass \, Fraction_{water}} = \frac{0.004 \, g}{0.016} = 0.25 \, g \]This calculation was key to solve further parts of the exercise regarding gas pressure and volume.

Gas Laws

Gas laws describe the behavior of gases and relate variables like pressure, volume, and temperature. Important gas laws include:

• Boyle's Law: At constant temperature, the pressure and volume of a gas are inversely proportional.
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.
• Gay-Lussac's Law: At constant volume, the pressure of a gas is directly proportional to its temperature in Kelvin.

These laws can be combined to form the Ideal Gas Law. Each plays a role in understanding the conditions inside the popcorn kernel before it pops. In our exercise, we heavily relied on the Ideal Gas Law to navigate through different gas conditions and calculated the necessary values.