A Rigid Plastic Container Holds 1.00 L Of Methane Gas At 0.868 Atm. How Much Pressure Will The Gas Exert If The Temperature Is Raised To $44.6^{\circ} C$?Before Solving:a. Which Variables Are Changing?b. What Is The Relationship Between

Mar 13, 2025 by ADMIN 239 views

$A rigid plastic container holds 1.00 L of methane gas at 0.868 atm. How much pressure will the gas exert if the temperature is raised to $44.6^{\circ} C$?$

Problem Analysis

To solve this problem, we need to understand the relationship between pressure, volume, and temperature of a gas. The ideal gas law is given by the equation:

PV = nRT

where:

P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the gas constant
T is the temperature of the gas in Kelvin

In this problem, we are given the initial pressure (0.868 atm), the initial volume (1.00 L), and the final temperature (44.6°C). We need to find the final pressure of the gas.

Variables Changing

The variables that are changing in this problem are:

Temperature (T): The temperature is raised from an unknown value to 44.6°C.
Pressure (P): We need to find the final pressure of the gas.
Volume (V): The volume is constant, as the container is rigid.

Relationship between Variables

The ideal gas law shows that the pressure of a gas is directly proportional to the temperature, provided that the volume and the number of moles are constant. Mathematically, this can be expressed as:

P ∝ T

This relationship can be used to solve the problem.

Solution

To solve the problem, we can use the following steps:

Convert the final temperature from Celsius to Kelvin:

T (K) = T (°C) + 273.15 T (K) = 44.6 + 273.15 = 317.75 K

Use the ideal gas law to find the final pressure:

P1 / T1 = P2 / T2

where:

P1 is the initial pressure (0.868 atm)
T1 is the initial temperature (unknown)
P2 is the final pressure (unknown)
T2 is the final temperature (317.75 K)

Rearrange the equation to solve for P2:

P2 = P1 × T2 / T1

However, we do not know the initial temperature. To find the initial temperature, we can use the ideal gas law:

P1 × V1 / T1 = P2 × V2 / T2

Since the volume is constant, we can simplify the equation:

P1 / T1 = P2 / T2

We can now substitute the values:

0.868 atm / T1 = P2 / 317.75 K

To find the initial temperature, we can use the ideal gas law:

P1 × V1 / T1 = nR

We can rearrange the equation to solve for T1:

T1 = P1 × V1 / (nR)

However, we do not know the number of moles (n). To find the number of moles, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n:

n = PV / (RT)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T

Q&A

Q: What is the ideal gas law?

A: The ideal gas law is a mathematical equation that describes the behavior of ideal gases. It is given by the equation:

PV = nRT

where:

P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the gas constant
T is the temperature of the gas in Kelvin

Q: What is the relationship between pressure and temperature?

A: The ideal gas law shows that the pressure of a gas is directly proportional to the temperature, provided that the volume and the number of moles are constant. Mathematically, this can be expressed as:

P ∝ T

Q: How can we solve the problem using the ideal gas law?

A: To solve the problem, we can use the following steps:

Convert the final temperature from Celsius to Kelvin:

T (K) = T (°C) + 273.15 T (K) = 44.6 + 273.15 = 317.75 K

Use the ideal gas law to find the final pressure:

P1 / T1 = P2 / T2

where:

P1 is the initial pressure (0.868 atm)
T1 is the initial temperature (unknown)
P2 is the final pressure (unknown)
T2 is the final temperature (317.75 K)

Rearrange the equation to solve for P2:

P2 = P1 × T2 / T1

However, we do not know the initial temperature. To find the initial temperature, we can use the ideal gas law:

P1 × V1 / T1 = P2 × V2 / T2

Since the volume is constant, we can simplify the equation:

P1 / T1 = P2 / T2

We can now substitute the values:

0.868 atm / T1 = P2 / 317.75 K

Q: How can we find the initial temperature?

A: To find the initial temperature, we can use the ideal gas law:

P1 × V1 / T1 = nR

We can rearrange the equation to solve for T1:

T1 = P1 × V1 / (nR)

However, we do not know the number of moles (n). To find the number of moles, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n:

n = PV / (RT)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n) or the temperature (T). To find the number of moles and the temperature, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for n and T:

n = PV / (RT) T = PV / (nR)

However, we do not know the gas constant (R). To find the gas constant, we can use the ideal gas law:

PV = nRT

We can rearrange the equation to solve for R:

R = PV / (nT)

However, we do not know the number of moles (n)