# Does Molecular Weight of a Gas affect its lifting properties at the same velocity over the same wing?

Let's say we test the coefficient of lift of the same wing at the same AOA and the same flow velocity at the same pressure.

The difference is wind tunnel A will contain helium (molecular weight 4) gas and wind tunnel B will contain air (average molecular weight around 29).

Which test will produce a higher lift coefficient? Why?

The wing will produce much less lift in helium than in air; all else being equal, lift is proportional to the molar mass of the gas in question.

The lift equation states that lift $$L$$ is proportional to the air density $$\rho$$ (rho):

$$L = \frac12 \rho \ v^2 \ S \ C_L,$$

where $$v$$ is the true airspeed, $$S$$ is the wing area, and $$C_L$$ is the coefficient of lift.

Meanwhile, one form of the ideal gas law tells us how the air density $$\rho$$ is related to the molar mass $$M$$:

$$\rho = \frac{MP}{RT},$$

where $$P$$ is the (static) pressure, $$R$$ is the gas constant, and $$T$$ is the absolute temperature.

Combining both of these equations, we find that

$$L = \frac{M P \ v^2 \ S \ C_L}{2 R T}.$$

In words, lift is

• the molar mass of the gas,
• times the static pressure,
• times the square of the airspeed,
• times the wing area,
• times the coefficient of lift,
• divided by the absolute temperature,
• divided by a constant (the gas constant $$R$$),
• divided by $$2$$.