# Virtual Temperature Calculator

Created by Michael Darcy
Reviewed by Dominik Czernia, PhD and Alexa Ruiz
Last updated: Oct 09, 2022

Our virtual temperature calculator is designed to help you find one of the many essential variables used in computer games atmospheric thermodynamics. If the name seems a bit confusing or you need a little refresher, read on!

In this article, you'll learn the virtual temperature definition, how it's related to the density of moist air, and how to use the virtual temperature formula. As the cherry on the top, we'll also present its applications - tropical cyclones included!

## What are we calculating? Virtual temperature definition

Although its name may suggest otherwise, the virtual temperature is a very real concept. Well, you can't measure it directly using a thermometer, but it's relatively easy to find using an appropriate formula (even easier with the virtual temperature calculator). So, what is it? The virtual temperature is the temperature a dry air parcel would need to have to have the same density as the moist air, provided that they had equal volume and pressure.

Density measures the unit of mass per volume, and because molecules of elements have different masses, changes in composition affect the air parcel density. Typically, the molecular weight of humid air is lower than that of dry air.

## Why is virtual temperature always warmer than temperature?

The density of moist air is lower than that of dry air. Since we don't want to change its composition to make these values equal, we can warm up the dry air to make it less dense. This makes the virtual temperature higher than the measured temperature.

To make this quantitative and not just make-believe, here are some numbers. Dry air is composed mainly of nitrogen (78%) and oxygen (21%), whose molecular weights are 28 g/mol and 32 g/mol, respectively. If we add moisture, some of these molecules are replaced by water with a molecular weight of only 18 g/mol. Therefore, the overall molecular weight of humid air is lower. To match its density, we can increase the kinetic energy of dry air parcel molecules by raising the temperature, and spaces between them. We find the value of this temperature using the virtual temperature equation.

## How to find the virtual temperature? The virtual temperature calculator

The virtual temperature calculator uses two methods to obtain the result, so we'll cover both of them:

1. The first virtual temperature formula is , also known as the ideal gas law - this time, for moist air. Several transformations eventually yield:
$T_v = T(1 + 0.61w),$

where:

• $T_v$ - Virtual temperature, in K;

• $T$ - Air temperature, in K; and

• $w$ - Mixing ratio, in kg/kg.

This formula looks simple, which is nice, but it requires additional calculations to find $w$. Therefore, sometimes it may be helpful to use another equation; or the mixing ratio of air calculator.

1. Another answer to "how to find the virtual temperature?" is the following formula:
$T_v = \frac{T}{1 - 0.379 \frac{e}{p_{\text{station}}}}$

where:

• $p_{\text{station}}$ - Station air pressure, in mb. In this case, a station is simply your location above sea level.
• $e$ - Actual vapor pressure. Our virtual temperature calculator computes it for you (you can see it in the advanced mode at the bottom), but here's the formula:
$e = 6.11 \times 10^{7.5 \times \frac{T_{\text{dew point}}}{237.7 + T_{\text{dew point}}}}$

where: $T_{\text{dew point}}$ is the dew point temperature in Celcius. If you are wondering how to calculate the dew point, check our dew point calculator.

## Why is the density temperature correction important?

Neither the virtual temperature definition nor relevant formulas immediately explain where such quantity would be applicable. Well, it's mainly used in meteorology and atmospheric thermodynamics for various purposes:

• Computing convective available potential energy (CAPE), which measures how much energy an air parcel would gain by being raised to a specific height in the atmosphere. It's expressed as the potential energy per kilogram of air mass (which you can read more about in our potential energy calculator). Including the density temperature correction
• Since virtual temperature allows us to accurately compute CAPE, it also indirectly helps Here, by storm, we mean not only regular thunderstorms but also tornadoes and cyclones.
• Density temperature is also useful in simplifying the calculations. You don't need to use a particular variant of the equation of state for air when humidity is present. Instead, you can substitute the virtual temperature for the air temperature in the ideal gas law.
• Another use in meteorology is the hypsometric equation, which solves for thickness (vertical distance) between two pressure levels in the atmosphere. Moreover, it shows that the thickness is proportional to the mean virtual temperature of the layer, which is considered to be a powerful statement.

As you can see, although perhaps non-descriptive at first, this quantity plays quite a significant role in many areas of science, and knowing how to find the virtual temperature can indeed be helpful.

## Using the virtual temperature formula in practice

Nothing can beat a good, old example when explaining something scientific. To make it more realistic, let's use actual data from New Orleans, Louisiana, and apply the virtual temperature equation.

On a certain day, the air temperature was 91 °F (32.8 °C) with a dew point of 74 °F (23.3 °C) and pressure of 30.04 in Hg (1017.27 hPa). Since Louisiana lies in the famous Tornado Alley, calculating CAPE and trying to predict the strength of a potential storm would be nice, but it's a little bit out of our scope today. However, we can make the first step!

Inputting this data into the virtual temperature calculator gives us the result of 96.95 °F (36.08 °C).

## FAQ

### How to find the virtual temperature with hypsometric equation?

To calculate the mean virtual temperature with the hypsometric equation, you have to rearrange it to the following form:

$T_v = g \frac{z_2 - z_1}{R_d \ln(\frac{p_1}{p_2})}$

where

• $g$ - Gravitational constant;
• $p_2, \ p_1$ - Upper and lower pressure levels;
• $z_2, \ z_1$ - Upper and lower heights the pressure levels are at;
• $R_d$ - Dry air gas constant; and
• $\ln$ - Natural logarithm.

### Why to use virtual temperature for CAPE?

Using virtual temperature for CAPE calculations allows for more accurate results. Research has shown that, especially for smaller values, neglecting the density temperature correction can yield a substantial relative error, and making corrections isn't the simplest task.

### What is the virtual temperature of air with temperature of 20 °C and mixing ratio of 6.70 g/kg?

The virtual temperature for these conditions is 294.34 K or 21.19 °C.

Let's take a look at the equation and substitute the values. We've already converted the temperature to Kelvin and the mixing ratio from g/kg to kg/kg. This gives:

Tᵥ = 293.15 * (1 + 0.61 * 0.001 * 6.70) = 294.34 K.

Michael Darcy
Method
Use temperatures
Air temperature
°F
Dew point
°F
Station pressure
psi
Virtual temperature
°F
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