Question

At what depth will ascending mantle rock with a temperature of \(1600 \mathrm{~K}\) melt if the equation for the solidus temperature \(T\) is $$ T(K)=1500+0.12 p(\mathrm{MPa}) $$ Assume \(\rho=3300 \mathrm{~kg} \mathrm{~m}^{-3}, g=10 \mathrm{~m} \mathrm{~s}^{-2},\) and the mantle rock ascends at constant temperature.

Short answer

The mantle rock will melt at a depth of approximately 25,252 meters.

Step-by-step solution

Understand the Solidus Equation

The solidus temperature equation given is \( T(K) = 1500 + 0.12p(\text{MPa}) \). This means the temperature at which rock begins to melt depends linearly on pressure \( p \) in megapascals (MPa). We need to find the depth at which the mantle rock at 1600 K starts to melt.

Set Temperature Equal to given Temperature

Set the solidus temperature equation to 1600 K because we need to find when the temperature of the rock matches the solidus temperature. Thus, \( 1500 + 0.12p = 1600 \).

Solve for Pressure

Subtract 1500 from both sides of the equation to isolate the pressure term: \( 0.12p = 100 \). Divide both sides by 0.12 to solve for \( p \): \( p = \frac{100}{0.12} = 833.33 \) MPa.

Calculate Pressure in Pascals

Convert the pressure from MPa to Pa because 1 MPa = \(10^6\) Pa. Therefore, \( p = 833.33 \times 10^6 \) Pa.

Use Pressure to Find Depth

The pressure exerted by a column of rock is given by \( p = \rho g h \), where \( \rho = 3300 \text{ kg/m}^3 \), \( g = 10 \text{ m/s}^2 \), and \( h \) is the depth. Substituting the values, we have \( 833.33 \times 10^6 = 3300 \times 10 \times h \).

Solve for Depth

Rearrange the equation to solve for \( h \): \( h = \frac{833.33 \times 10^6}{3300 \times 10} \). Simplifying, \( h \approx 25,252 \text{ m} \), which is the depth where the rock will begin to melt.

Key concepts

Solidus Temperature

The solidus temperature is a critical concept in geology. It is the temperature at which a rock starts to melt, particularly a mantle rock in this context. As the rock ascends within the Earth, the temperature at which it begins to melt will depend on the surrounding pressure. This relationship is described by the solidus temperature equation.
The equation given in the exercise is:

• \( T(K) = 1500 + 0.12p(\mathrm{MPa}) \) ,

where \( T \) is the solidus temperature in Kelvin, and \( p \) is the pressure in megapascals (MPa).
This equation shows a linear relationship between pressure and the solidus temperature, meaning as pressure increases, the temperature required to start melting also increases. Understanding this concept is key to solving the exercise, as it helps determine the conditions under which mantle rock begins to melt as it ascends.

Mantle Rock

Mantle rocks make up a large part of the Earth's interior. These rocks are mostly silicate minerals, making them dense and heavy.
In our exercise, we're looking at the mantle rock ascending within the Earth, maintaining a constant temperature of 1600 K.
The density of mantle rock, given as \( \rho = 3300 \mathrm{kg} \mathrm{m}^{-3} \), plays a role in calculating the pressure it exerts at a given depth.
Knowing the properties of mantle rocks is essential to understand geological processes, such as volcanic activity and plate tectonics. As these rocks move upwards towards the surface, the change in pressure and temperature can result in partial melting, which can lead to magmatic processes.

Pressure Conversion

In geology, it's common to work with pressure as it significantly impacts geological states and processes. Pressure is often expressed in megapascals (MPa) but can be converted into units like pascals (Pa) for mathematical convenience.
In our step-by-step solution, we needed to convert megapascals to pascals because our equation dealing with depth uses pascals.
The conversion is straightforward:

• 1 MPa = \(10^6\) Pa.

We found that the pressure at which mantle rock reaches the solidus temperature is 833.33 MPa, which equals \(833.33 \times 10^6\) Pa.
Understanding how to convert between these units is crucial for calculating the stress and pressure on geological materials, providing insights into their behavior under different conditions.

Geological Depth Calculation

Calculating geological depth involves understanding how pressure is related to depth within the Earth. In our problem, we used the pressure to determine the depth at which the mantle rock begins to melt.
This is calculated by knowing:

• The density of the rock \( \rho = 3300 \mathrm{kg} \mathrm{m}^{-3} \)
• The acceleration due to gravity \( g = 10 \mathrm{m} \mathrm{s}^{-2} \)

The pressure exerted by a column of rock is given by the equation:

• \( p = \rho g h, \)

where \( h \) is the depth.
By rearranging this equation, \( h = \frac{p}{\rho g} \), we calculated the depth as approximately 25,252 meters.
This depth calculation helps geologists understand how deep they have to go before encountering conditions like those described in our exercise. Understanding these calculations enables us to predict at what depths certain geological phenomena, like mantle melting, might occur.