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# Using a freezing point depression osmometer to measure serum osmolality

## Problem

Osmolality is a measurement of the total solute concentrations dissolved in a solvent. An approximation of serum osmolality can be calculated using the following formula: 2, open bracket, N, a, plus, close bracket, plus, open bracket, G, l, u, c, o, s, e, close bracket, slash, 18, plus, open bracket, U, r, e, a, close bracket, slash, 2, point, 8
Serum osmolality can be measured more accurately in a laboratory setting by using a device called a freezing point depression osmometer (Figure 1).
Figure 1: Schematic Diagram of a Freezing Point Depression Osmometer
In a freezing point depression osmometer, a sample of serum is introduced to a cooling chamber where it is cooled to a temperature below its expected freezing point. The sample is stirred with a wire, which causes the sample to become partially crystallized. The heat generated from this process brings the sample to a temperature that is equal to its freezing point. A standard freezing curve is generated, and the osmolality of the serum sample is calculated from the freezing point depression using the following equation: ∆, T, equals, k, start subscript, f, end subscript, times, o, s, m, o, l, a, l, i, t, y, comma where k, start subscript, f, end subscript is known as the cryoscopic constant, which is equal to 1, point, 86, K, \cdotp, k, g, slash, m, o, l, point
Figure 2 depicts a typical freezing point curve that is generated from a freezing point depression osmometer.
Figure 2: Standard Freezing Curve
The normal values of serum osmolality range from 280 to 300 m, O, s, m, slash, k, g of H, start subscript, 2, end subscript, O. A wide array of medical conditions can lead to abnormally high or low serum osmolality. Two endocrine diseases that can alter the serum osmolality are SIADH and Diabetes Mellitus. Syndrome of Inappropriate Antidiuretic Hormone Hyper-secretion (SIADH) is a condition where free water is abnormally retained by the kidneys. This syndrome is characterized by low serum sodium levels. Diabetes mellitus is a condition that occurs when the pancreas no longer produces enough insulin or when the body does not respond to the insulin that is produced by the pancreas. This causes the serum glucose to remain elevated.
A sample of serum has 140 mEq/dL of sodium, 90 mEq/dL of glucose, and 14 mEq/dL of urea. What is the osmolality of the sample?