A steel cylinder with rigid walls is evacuated to a high...

A steel cylinder with rigid walls is evacuated to a high degree of vacuum; you then put a small amount of helium into the cylinder. The cylinder has a pressure gauge that measures the pressure of the gas inside the cylinder. You place the cylinder in various temperature environments, wait for thermal equilibrium to be established, and then measure the pressure of the gas. You obtain these results: (a) Recall (Chapter 17 ) that absolute zero is the temperature at which the pressure of an ideal gas becomes zero. Use the data in the table to calculate the value of absolute zero in ${ }^{\circ} \mathrm{C}$. Assume that the pressure of the gas is low enough for it to be treated as an ideal gas, and ignore the change in volume of the cylinder as its temperature is changed. (b) Use the coefficient of volume expansion for steel in Table 17.2 to calculate the percentage change in the volume of the cylinder between the lowest and highest temperatures in the table. Is it accurate to ignore the volume change of the cylinder as the temperature changes? Justify your answer.

A steel cylinder with rigid walls is evacuated to a high degree of vacuum; you then put a small amount of helium into the cylinder. The cylinder has a pressure gauge that measures the pressure of the gas inside the cylinder. You place the cylinder in various temperature environments, wait for thermal equilibrium to be established, and then measure the pressure of the gas. You obtain these results:

(a) Recall (Chapter 17 ) that absolute zero is the temperature at which the pressure of an ideal gas becomes zero. Use the data in the table to calculate the value of absolute zero in ${ }^{\circ} \mathrm{C}$. Assume that the pressure of the gas is low enough for it to be treated as an ideal gas, and ignore the change in volume of the cylinder as its temperature is changed. (b) Use the coefficient of volume expansion for steel in Table 17.2 to calculate the percentage change in the volume of the cylinder between the lowest and highest temperatures in the table. Is it accurate to ignore the volume change of the cylinder as the temperature changes? Justify your answer.

Key Concepts

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a material-specific property that quantifies the rate at which a material's dimensions change per degree change in temperature. This concept is critical when evaluating how much structural materials, such as metal cylinders, expand or contract with temperature, which in turn can influence the conditions under which gas laws are applied.

Thermal Expansion

Thermal expansion is the tendency of materials to change their dimensions in response to changes in temperature. This principle is important in experiments where the physical dimensions of a container might vary with temperature, potentially affecting the internal volume and the pressure measurements of the gas inside.

Ideal Gas Law

The ideal gas law is a fundamental principle in thermodynamics that relates pressure, volume, temperature, and the number of moles of a gas. It is particularly useful for understanding how the pressure of a gas changes with temperature when volume remains constant, as it assumes that gas particles move randomly and do not interact except through elastic collisions.

Absolute Zero

Absolute zero is the theoretical temperature at which all thermal motion of particles ceases and, for an ideal gas, the pressure would drop to zero. It serves as the lower limit of the thermodynamic temperature scale and is a crucial point in understanding the relationship between temperature and molecular activity.

Transcript

00:01 So we're trying to find, they're giving us temperatures of this evacuated cylinder and pressures of this evacuated cylinder.

00:11 They're giving us the fact that the cylinder is steel.

00:16 So we have steel cylinder evacuated.

00:25 And then it has helium, a small amount of helium.

00:36 Then they're giving us a table.

00:38 So we have the table of t degrees.

00:43 Celsius on the x -axis and then p pressure and pascal is on the y axis and we have some values so we have 195 negative 190 .8 254 0 .0 980 33 .3 .3 900 999199 100 100 and 1214 and then we have 23 32 degrees celsius and 6 ,165 pascal.

01:29 So using technology, i would say use ti 84, t8589, or excel.

01:42 Find the y equals mxpsb.

01:45 Find this.

01:46 So we're trying to find the slope intercept.

01:49 And when we model this, we find that the slope is going to be equal to 3 .230.

01:58 This would be in terms of pascals per degree celsius.

02:04 And then b would equal the y intercept.

02:11 And this would be equal to 888 .895858 .858 .858.

02:19 So that means that the pressure is going to be equal, rather the pressure y, can be modeled as 3 .232.

02:35 Per degree celsius times t, which would be our x, and then plus b, 888 .8958 .898 .858.

02:47 Pascal's...