# Solving the Perfect Gas Equation for Volume at Sea Level and 2000m

• Benny
In summary, This conversation discusses the process of finding the volume of a balloon filled with hydrogen at different altitudes and temperatures. The perfect gas equation and adiabatic condition are used to determine the volume in different scenarios. The adiabatic condition is used when there is no heat flow into or out of the balloon, while the perfect gas equation is not applicable in this case as the temperature of the balloon is unknown. The adiabatic condition is also used to take into account the work done by the gas on the atmosphere, causing the gas to cool. Clarification is also provided on properly identifying the system to which the equations are being applied.
Benny
Hi, I'm not sure how to do the following. Could someone help me out?

Q. At sea level, where the barometric pressure is 101.3 kN/m^2 and the temperature is 20 degrees celcius, a balloon is filled with 2.5 kg of hydrogen also at 20 degrees celcius.

(i) Find the volume of the balloon at sea level.

(ii) If the balloon rises quickly to an altitude of 2000m where the barometric pressure is 80 kN/m^2 and the temperature is 2.0 degrees celcius find its volume.

(iii) If the balloon stays at 2000 m altitude until its temperature stabilises at 2 degrees celcius find its volume.

The pressure difference between the hydrogen inside the balloon and the air outside may be neglected.

My working:

(i) I just used the perfect gas equation as follows.

$$V_{H_2 } = \frac{{m_{H_2 } R_{H_2 } T}}{P}$$

Is that right?

(ii) For this part I used:

$$\frac{{P_1 V_1 }}{{T_1 }} = \frac{{P_2 V_2 }}{{T_2 }}$$

where the values of symbols with 1 (one) as the subscript are the same as those given or used in part (i) while P_2 = 80 kN/m^2 and T_2 = 295 K.

Solve for V_2 to find the volume of the balloon?

(iii) For this part I essentially just used $$\frac{{P_1 V_1 }}{{T_1 }} = \frac{{P_2 V_2 }}{{T_2 }}$$ again. However, this time wouldn't P_1 = P_2 = 80 kN/m^2 since the barometric pressure hasn't changed from part (ii)? Also, the temperature "stabilises at 2 degrees celcius" which is the same temperature as in part (ii). But this seems to imply that the required volume is just the same as that found in part (ii).

I'm not really sure what to do here. Can someone check my working and point me in the right direction? Thanks.

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Benny said:
(ii) If the balloon rises quickly to an altitude of 2000m where the barometric pressure is 80 kN/m^2 and the temperature is 2.0 degrees celcius find its volume.

(ii) For this part I used:

$$\frac{{P_1 V_1 }}{{T_1 }} = \frac{{P_2 V_2 }}{{T_2 }}$$

where the values of symbols with 1 (one) as the subscript are the same as those given or used in part (i) while P_2 = 80 kN/m^2 and T_2 = 295 K.

Solve for V_2 to find the volume of the balloon?
Since the balloon rises quickly, the temperature of the balloon does not have time to change to the surrounding temperature. In other words, there is no heat flow into or out of the balloon, so the expansion is adiabatic.

$$PV^\gamma = K$$

to determine the final volume. For air, being mostly diatomic, $\gamma = 1.4$.

AM

Thanks for explaining part (ii). I can accept that the final volume can be found from one of the 'adiabatic versions' of the gas equation but I would like to know why the equation I originally used in my opening post, is not valid in this case. I've read through my notes but I still don't know when the equation I originally used is applicable.

Benny said:
Thanks for explaining part (ii). I can accept that the final volume can be found from one of the 'adiabatic versions' of the gas equation but I would like to know why the equation I originally used in my opening post, is not valid in this case. I've read through my notes but I still don't know when the equation I originally used is applicable.

Quick or insulated things generally follw the adiabatic condition - no heat exchange between the gas and the outside world.

Benny said:
Thanks for explaining part (ii). I can accept that the final volume can be found from one of the 'adiabatic versions' of the gas equation but I would like to know why the equation I originally used in my opening post, is not valid in this case. I've read through my notes but I still don't know when the equation I originally used is applicable.
The short answer is that you do not know what the temperature of the balloon is so you can't use PV=nRT to find the volume.

As the balloon rises quickly and the ambient pressure decreases, the gas in the balloon expands and does work on the atmosphere. This causes the hydrogen gas to lose internal energy (since there is no heat being added, the energy required to perform that work must come from internal energy of the gas) so the gas cools. The relationship between Pressure and Volume is given by the adiabatic condition. Since P = nRT/V, the adiabatic condition can also be expressed as:

$$TV^\gamma/V = TV^{(\gamma -1)} = K/nR = \text{Constant}$$

AM

I think I needed to properly identify the system to which I'm applying the equations, but when I tried this question I used information about the surroundings which don't apply to the hygrogen in the balloon. Thanks for clearing that up.

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## What is the perfect gas equation for calculating volume at sea level and 2000m?

The perfect gas equation is PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the temperature.

## How do you solve for volume using the perfect gas equation?

To solve for volume, you will need to know the pressure, number of moles, gas constant, and temperature. Plug these values into the equation and solve for V.

## Why is it important to consider sea level and 2000m when solving the perfect gas equation for volume?

The pressure and temperature at different altitudes affect the volume of a gas. At higher altitudes, the pressure is lower and the temperature is cooler, which can result in a smaller volume of gas.

## Can the perfect gas equation be used for all gases?

The perfect gas equation can be used for ideal gases, which follow certain assumptions such as having no intermolecular forces and occupying negligible volume. Real gases may deviate from the ideal gas law at high pressures and low temperatures.

## What are some potential sources of error when using the perfect gas equation to calculate volume?

Potential sources of error can include inaccurate measurement of pressure or temperature, deviations from ideal gas behavior, and variations in atmospheric conditions at different altitudes.

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