On a day when atmospheric pressure if 76cmHg, the pressure gauge on the tank reads the pressure inside to be 400cmHg. the gas in the tank has a temperature of 9 degrees celcuis. If the tank is heated to 31 degrees celcuis by the sun, and no gas exists, what will the pressure gauge read?

P V = n R T

P/T = n R/V = constant here
T1 = 273 + 9
T2 = 273 + 31

P1 = 76 + 400
P2 = 76 + x where x is new gage pressure

P2/T2 = P1/T1
so P2 = (T2/T1) P1
x = P2 - 76

To find the new pressure reading on the gauge after the tank is heated, we need to apply the ideal gas law, which states:

PV = nRT

Where:
P = Pressure
V = Volume
n = Number of moles of gas
R = Ideal gas constant
T = Temperature in Kelvin

Let's solve this step-by-step:

Step 1: Convert the temperatures to Kelvin.
Given:
Initial temperature (T1) = 9 degrees Celsius
Final temperature (T2) = 31 degrees Celsius

To convert Celsius to Kelvin, we use the formula:
T(K) = T(°C) + 273.15

T1(K) = 9 + 273.15
T2(K) = 31 + 273.15

T1(K) = 282.15 K
T2(K) = 304.15 K

Step 2: Calculate the initial pressure (P1) using the given atmospheric pressure and the pressure reading on the tank.
Given:
Atmospheric pressure (P_atm) = 76 cmHg
Pressure reading on the tank (P_gauge) = 400 cmHg

P1 = P_gauge - P_atm

P1 = 400 cmHg - 76 cmHg
P1 = 324 cmHg

Step 3: Calculate the final pressure (P2) using the ideal gas law.
Given:
Initial pressure (P1) = 324 cmHg
Initial temperature (T1) = 282.15 K
Final temperature (T2) = 304.15 K

Using PV = nRT, we can rewrite it as P1V1/T1 = P2V2/T2

Since the tank is empty (no gas exists - V2 = 0), we have:
P1/T1 = P2/0

Since division by zero is undefined, we need to consider an alternative approach.
If the tank is heated, the volume will increase, and we can assume the pressure inside will remain constant. Therefore, the pressure gauge will still read P1.

Therefore, the pressure gauge will read 324 cmHg after the tank is heated.

To determine the pressure the gauge will read after the tank is heated, we can use the combined gas law equation, which relates the initial and final conditions of a gas when temperature, pressure, and volume change. The combined gas law equation is as follows:

(P1 * V1) / T1 = (P2 * V2) / T2

Where:
P1 = Initial pressure
V1 = Initial volume (assumed constant)
T1 = Initial temperature
P2 = Final pressure (what we need to find)
V2 = Final volume (assumed constant)
T2 = Final temperature

Let's find each of these variables step by step:

1. Convert the atmospheric pressure of 76 cmHg into the SI unit of pressure, which is Pascal (Pa). 1 cmHg is approximately equal to 133.3224 Pa. Therefore:

P1 = 76 cmHg * 133.3224 Pa/cmHg = 10132.32 Pa

2. Convert the pressure inside the tank to Pascal:

P2 = 400 cmHg * 133.3224 Pa/cmHg = 53329.28 Pa

3. Convert the initial and final temperatures from Celsius to Kelvin. Kelvin is the absolute temperature scale and is obtained by adding 273.15 to the Celsius temperature:

T1 = 9°C + 273.15 = 282.15 K
T2 = 31°C + 273.15 = 304.15 K

4. Since the tank does not have an outlet for the gas to escape, we assume the volume remains constant. Thus:

V1 = V2

Now, we can rearrange and substitute all the known values into the combined gas law equation:

(P1 * V1) / T1 = (P2 * V2) / T2

Since V1 = V2, we can simplify the equation to:

P1 / T1 = P2 / T2

Substituting the values we obtained earlier:

(10132.32 Pa) / (282.15 K) = P2 / (304.15 K)

Solving for P2, we find:

P2 = (10132.32 Pa) * (304.15 K) / (282.15 K) ≈ 10955.21 Pa

To convert this pressure back to cmHg, we divide by the conversion factor:

(10955.21 Pa) / 133.3224 Pa/cmHg ≈ 82.20 cmHg

Therefore, the pressure gauge will read approximately 82.20 cmHg when the tank is heated to 31 degrees Celsius.